1 Exterior differential systems
2 Tableaux
3 Example: almost complex structures
4 Prolongation
5 Cartan's test
6 The characteristic variety
7 Proof of the Cartan–Kaehler theorem
8 Cauchy characteristics
9 Example: conformal maps of surfaces
10 Example: Weingarten surfaces
A The Cauchy–Kovalevskaya theorem
B Symbol and characteristic variety
C Analytic convergence
D The moving frame
E Curvature of surfaces
F The Gauss–Bonnet theorem
G Geodesics on surfaces
Hints
Bibliography

##### Citation preview

Benjamin McKay

Introduction to Exterior Differential Systems

April 11, 2021

iii

Preface

To the reader who wants to dip a toe in the water: just read chapter 1. The reader who continues on to chapters 2 and 4 will pick up the rest of the tools. Subsequent chapters prove the theorems. We assume that the reader is familiar with elementary differential geometry on manifolds and with differential forms. These lectures explain how to apply the Cartan–Kähler theorem to problems in differential geometry. Given some differential equations, we want to decide if they are locally solvable. The Cartan–Kähler theorem gives a linear algebra test: if the equations pass the test, they are locally solvable. We give the necessary background on partial differential equations in appendices A, B, and the (not so necessary) background on moving frames in appendices D, F, G. The reader should be aware of [4], which we will follow closely, and also the canonical reference work [3] and the canonical textbook [19]. I wrote these notes for lectures at the Banach Center in Warsaw, at the University of Paris Sud, and at the University of Rome Tor Vergata. I thank Jan Gutt, Gianni Manno, Giovanni Moreno, Nefton Pali, and Filippo Bracci for invitations to give those lectures, and Francesco Russo for the opportunity to write these notes at the University of Catania.

v

Contents

1

Exterior differential systems

2

Tableaux

3

Example: almost complex structures

4

Prolongation

33

5

Cartan’s test

39

6

The characteristic variety

7

Proof of the Cartan–Kähler theorem

8

Cauchy characteristics

9

Example: conformal maps of surfaces

1

13

47

B Symbol and characteristic variety D The moving frame

97

101

E

Curvature of surfaces

F

The Gauss–Bonnet theorem

G Geodesics on surfaces Hints

149

Bibliography Index

181

185

vi

115 133

69

73

A The Cauchy–Kovalevskaya theorem Analytic convergence

53

59

10 Example: Weingarten surfaces

C

25

125

77 87

Chapter 1

Exterior differential systems

We define exterior differential systems, and state the Cartan–Kähler theorem.

Background material from differential geometry By the expression analytic, we mean real analytic. We assume that all of our manifolds, maps, vector fields and differential forms are analytic. (In a few exercises, we describe some such thing as smooth, i.e. C ∞ , not necessarily analytic.) A submanifold is an immersion of manifolds, defined up to diffeomorphism of the domain. Denote the dimension of a manifold M by dim M . The codimension of a submanifold S of M is dim M − dim S. A hypersurface in a manifold M is a submanifold of codimension one. A hyperplane in a vector space V is a linear subspace W ⊂ V so that dim(V /W ) = 1.

Differential equations encoded in differential forms  To express a differential equation 0 = f x, u, ∂u ∂x , add a variable p to represent .. .. the derivative ∂u ∂x . Let ϑ = du − p dx, ω = dx. Let M be the manifold . M .= { (x, u, p) | 0 = f (x, u, p) } (assuming it is a manifold). Any submanifold of M on which 0 = ϑ and 0 6= ω is locally the graph of a solution. It is easy to generalize this to any number of variables and any number of equations of any order.

Exterior differential systems An integral manifold of a collection of differential forms is a submanifold on which the forms vanish. An exterior differential system on a manifold M is an ideal I ⊂ Ω ∗ of differential forms, closed under exterior derivative, splitting into a direct sum I = I 1 ⊕ · · · ⊕ I dim M of forms of various degrees: I k ..= I ∩ Ω k . Any collection of differential forms has the same integral manifolds as the exterior differential system it generates. 1

2

Exterior differential systems

Some trivial examples: the exterior differential system generated by a. 0, b. Ω ∗ , c. the pullbacks of all forms via a submersion, d. dx1 ∧ dy 1 + dx2 ∧ dy 2 in R4 , e. dy − z dx on R3 . 1.1 What are the integral manifolds of our trivial examples? By definition I 0 = 0, i.e. there are no nonzero functions in I. If we wish some functions to vanish, we can replace our manifold M by the zero locus of those functions (which might not be a manifold, a technical detail we will ignore).

Integral elements An integral element of an exterior differential system I is a linear subspace of a tangent space, on which all forms in I vanish. Every tangent space of any integral manifold is an integral element, but some integral elements of some exterior differential systems don’t arise as tangent spaces of integral manifolds. A 1-dimensional integral element is an integral line. A 2-dimensional integral element is an integral plane. 1.2 What are the integral elements of our trivial examples? 1.3 Suppose that the 1-forms in an exterior differential system span a subspace of constant rank in each cotangent space. Prove that there is an integral curve tangent to each integral line. 1.4 Give an example of an embedded integral manifold, whose every tangent space is a hyperplane in a unique integral element, but which is not a hypersurface in an integral manifold. 1.5 Prove that a k-dimensional linear subspace of a tangent space is an integral element of an exterior differential system I just when all k-forms in I vanish on it. The polar equations of an integral element E are the linear functions w ∈ Tm M 7→ ϑ(w, e1 , e2 , . . . , ek ) where ϑ ∈ I k+1 and e1 , e2 , . . . , ek ∈ E, for any k. They vanish on a vector w just when the span of { w } ∪ E is an integral element. The set of polar

Integral elements as points of the Grassmann bundle ∗ equations of E is a linear subspace of Tm M . If an integral element E lies in a larger one F , then all polar equations of E occur among those of F : larger integral elements have more (or at least the same) polar equations.

1.6 What are the polar equations of the integral elements of our trivial examples? An integral flag of an exterior differential system is a sequence of nested integral elements 0 = E0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ Ep with dimensions 0, 1, 2, . . . , p. Danger: most authors require that a flag have subspaces of all dimensions; we don’t: we only require that the subspaces have all dimensions 0, 1, 2, . . . , p up to some dimension p. Successive subspaces have successively larger space of polar equations. The characters s0 , s1 , . . . , sp of the flag are the increments in rank of polar equations: Ek has polar equations of rank s0 + s1 + · · · + sk . Consider all flags inside a given Ep . Polar equations remain linearly independent under small motions of a flag. So if a flag in Ep has maximal dimensional polar equations, i.e. character sums s0 , s0 + s1 , . . . , s0 + · · · + sp , then so do all nearby flags in Ep . The characters of the integral element Ep are those of any such flag. 1.7 What are the characters of the integral flags of our trivial examples?

Integral elements as points of the Grassmann bundle The rank p Grassmann bundle of a manifold M is the set of all p-dimensional linear subspaces of tangent spaces of M . 1.8 Recall how charts are defined on the Grassmann bundle. Prove that the Grassmann bundle is a fiber bundle. The integral elements of an exterior differential system I form a subset of the Grassmann bundle. We would like to see if that subset is a submanifold, and predict its dimension.

The Cartan–Kähler theorem An integral element, say of dimension p and with characters s0 , . . . , sp , predicts the dimension dim M + s1 + 2s2 + · · · + psp . It predicts correctly if the nearby integral elements form a submanifold of the Grassmann bundle of the predicted dimension. (We will see that they always sit in a submanifold of the predicted dimension.) An integral element which correctly predicts dimension is involutive. An exterior differential system on a manifold M is involutive if each component of M contains an involutive maximal integral element. (We will see that involutivity of an exterior differential system implies that involutive integral elements occur at a dense open subset of points of M .)

3

4

Exterior differential systems

Theorem 1.1 (Cartan–Kähler). Every involutive integral element of any analytic exterior differential system is tangent to an analytic integral manifold. 1.9 Give an example of an exterior differential system whose characters are not constant.

Example: Lagrangian submanifolds We employ the Cartan–Kähler theorem to prove the existence of Lagrangian submanifolds of complex Euclidean space. Let ϑ ..= dx1 ∧ dy 1 + dx2 ∧ dy 2 + · · · + dxn ∧ dy n . Let I be the exterior differential system generated by ϑ on M ..= R2n . Writing spans of vectors in angle brackets, Flag

Polar equations

E0 = { 0 } E1 = h∂x1 i .. .

{ 0 } dy 1 ..

.1 En = h∂x1 , ∂x2 , . . . , ∂xn i dy , dy 2 , . . . , dy n

Characters s0 = 0 s1 = 1 .. .

sn = 1

The flag predicts dim M + s1 + 2 s2 + · · · + n sn = 2n + 1 + 2 + · · · + n. The nearby integral elements at a given point of M are parameterized by dy = a dx, which we plug in to ϑ = 0 to see that a can be any symmetric matrix. So the space of integral elements has dimension dim M +

n(n + 1) n(n + 1) = 2n + , 2 2

correctly predicted. The Cartan–Kähler theorem proves the existence of Lagrangian submanifolds of complex Euclidean space, one (at least) through each point, tangent to each subspace dy = a dx, at least for any symmetric matrix a close to 0. 1.10 On a complex manifold M , take a Kähler form ϑ and a holomorphic volume form Ψ , i.e. closed forms expressed in local complex coordinates as √ −1 ϑ= gµ¯ν dz µ ∧ dz ν¯ , 2 Ψ = f (z) dz 1 ∧ dz 2 ∧ · · · ∧ dz n , with f (z) a holomorphic function and gµ¯ν a positive definite self-adjoint complex matrix of functions. Prove the existence of special Lagrangian submanifolds, i.e. integral manifolds of both ϑ and the imaginary part of Ψ .

5

The last character

The last character We always look for integral manifolds of a particular dimension p. For simplicity, we can assume that our exterior differential system contains all differential forms of degree p + 1 and higher. In particular, the p-dimensional integral elements are maximal. The polar equations of any maximal integral element Ep cut out precisely Ep , i.e. have rank dim M − p on Ep . We encounter s0 , s1 , . . . , sp polar equations at each increment, so rank s0 + s1 + · · · + sp . We can find sp from the other characters: s0 + s1 + · · · + sp−1 + sp = dim M − p. For even greater simplicity, we take this as a definition for the final character sp ; we only pretend that our exterior differential system contains all differential forms of degree p + 1 and higher.

Example: harmonic functions We will prove the existence of harmonic functions on the plane with given value and first derivatives at a given point. On M = R5x,y,u,ux ,uy , let I be the exterior differential system generated by ϑ ..= du − ux dx − uy dy, Θ ..= dux ∧ dy − duy ∧ dx. Note that

dϑ = dx ∧ dux + dy ∧ duy

also belongs to I because any exterior differential system is closed under exterior derivative. Any integral surface X on which 0 6= dx ∧ dy is locally the graph of ∂u a harmonic function u = u(x, y) and its derivatives ux = ∂u ∂x , ux = ∂x . Each integral plane E2 (i.e. integral element of dimension 2) on which dx ∧ dy 6= 0 is given by equations dux = uxx dx + uxy dy, duy = uyx dx + uyy dy, for a unique choice of 4 constants uxx , uxy , uyx , uyy subject to the 2 equations uxy = uyx and 0 = uxx +uyy . Hence integral planes at each point have dimension 2. The space of integral planes has dimension dim M + 2 = 5 + 2 = 7. Every integral line is the span E1 = hvi of a vector v = (x, ˙ y, ˙ ux x˙ + uy y, ˙ uxx x˙ + uxy y, ˙ uyx x˙ + uyy y) ˙ . Compute     x˙ dux + y˙ duy − (uxx x˙ + uxy y) ˙ dx − (uyx x˙ + uyy y) ˙ dy dϑ v = . Θ y˙ dux − x˙ duy + (uxx x˙ + uxy y) ˙ dy − (uyx x˙ + uyy y) ˙ dx

6

Exterior differential systems

and Flag

Polar equations

Characters

E0 ={ 0 } E1 =hvi

hϑi s0 = 1 hϑ, v dϑ, v Θi s1 = 2

We are only interested in finding integral surfaces; we compute the final character from s0 + s1 + s2 = dim M − 2. The characters are s0 , s1 , s2 = 1, 2, 0 with predicted dimension dim M + s1 + 2s2 = 5 + 2 + 2 · 0 = 7: involution. So harmonic functions exist near any point of the plane, with prescribed value and first derivatives at that point. 1.11 What are the integral elements and integral manifolds of the exterior differential system generated by du ∧ dx, dv ∧ dx, u du ∧ dv on R4x,y,u,v ?

Background material from differential geometry 1.12 Prove: constant rank linear equations, depending analytically on parameters, have a local solution depending analytically on those parameters. Lemma 1.2 (Cartan’s lemma). Suppose that ξ1 , ξ2 , . . . , ξk are linearly independent analytic 1-forms on some manifold M . (They need not form a basis.) Suppose that α1 , α2 , . . . , αk are analytic 1-forms satisfying 0 = α1 ∧ ξ1 + α2 ∧ ξ2 + · · · + αk ∧ ξk . Then therePis a unique symmetric matrix A = (aij ) of analytic functions so that αi = aij ξj . 1.13 Prove Cartan’s lemma using the solution of problem 1.12.

The Frobenius theorem The pullback of an exterior differential system by a map is the ideal generated by the pulled back forms. 1.14 Prove that the pullback of an exterior differential system is an exterior differential system. A exterior differential system is locally generated by forms with some property if every point lies in an open set so that the pullback to that open set is generated by forms with that property. Theorem 1.3 (Frobenius). Suppose that I is an exterior differential system on a manifold M . The following are equivalent; if any, hence all, hold we say that I is Frobenius. a. Every integral element of I lies in a unique p-dimensional integral element.

Example: surfaces with constant shape operator

b. There is a constant so that I has a p-dimensional integral element at each point of M , with character s0 equal to that constant, and all other characters s1 , s2 , . . . , sp vanishing. c. I has an involutive p-dimensional integral element at each point of M . I is locally generated by dim M − p linearly independent 1-forms. d. I is locally generated as an ideal of differential forms by dim M −p linearly independent 1-forms. e. I is locally by linearly independent 1-forms θ1 , θ2 , . . . , θq with P generated i i j dθ = − j ξj ∧ θ for some 1-forms ξji . f. I is locally generated by linearly independent 1-forms. The vector fields on which all 1-forms in I vanish are closed under Lie bracket. They span a p-dimensional linear subspace in each tangent space of M . g. Each point of M has coordinates x1 , . . . , xp , y 1 , . . . , y q so that I is locally generated by dy 1 , dy 2 , . . . , dy q . h. The p-dimensional I-integral manifolds form the leaves of a foliation F of M . I is locally generated by the 1-forms vanishing on the leaves. 1.15 Use a statement of the Frobenius theorem from a differential geometry textbook, or lemma 7.5 on page 55, to prove the Frobenius theorem. To solve both ux = f (x, y, u) and uy = g(x, y, u), note uxy = fy + fu uy = uyx = gx + gu ux , the compatibility condition between these equations. The exterior differential system generated by θ = du − f dx − g dy contains dθ = (fy + fu g − gx − gu f )dx ∧ dy + (fu dx + gu dy) ∧ θ, so takes the compatibility condition into account. By the Frobenius theorem, fu + fu g = gx + gu f everywhere just when there are solutions with arbitrary initial condition u(x0 , y0 ) = u0 .

Example: surfaces with constant shape operator In this section, we assume familiarity with appendix D. Use the Frobenius theorem: 1.16 Prove that, for point x ∈ E3 and plane P in E3 through x, any connected and embedded surface with zero shape operator passing through x and tangent to P at x is an open subset of P .

7

8

Exterior differential systems

1.17 An oriented surface is c0 -round, for a constant c0 > 0, if it has shape operator II(u, v) = c0 u · v at every point. Prove that, for each constant c0 ∈ R, point x ∈ E3 and oriented plane P through x, there is a c0 -round oriented surface passing through x, with oriented tangent space at x equal to P . If c0 6= 0, prove that any such surface, if connected and embedded, is an open subset of a sphere. 1.18 Prove that any connected and embedded surface in E3 with constant principal curvatures is an open subset of a plane, sphere or cylinder. 1.19 Prove that any connected embedded curve in E3 of constant curvature and torsion is an open subset of a line, circle or helix. 1.20 An umbilic point is a point of a surface in E3 where the shape operator is a multiple of the dot product: II(u, v) = c u · v, for some number c. Find all connected and embedded surfaces in E3 which are everywhere umbilic, with perhaps varying value of c. 1.21 Find every connected and embedded surface in E3 whose every geodesic is planar.

Background material from differential geometry A collection of 1-forms ω 1 , . . . , ω n on a manifold M coframes M , or is a coframe 1 n ∗ or coframing of M , if, at every point m ∈ M , ωm , . . . , ωm are a basis of Tm M.

Example: triply orthogonal webs A triply orthogonal web is a triple of foliations by surfaces whose leaves are pairwise perpendicular.

Images: Daniel Piker, 2015

Theorem 1.4. There are infinitely many triply orthogonal webs, depending on 3 functions of 2 variables, defined near any given point of 3-dimensional Euclidean space.

9

Example: triply orthogonal webs

Proof. Picture a triply orthogonal web. Each leaf is perpendicular to a unique unit length 1-form ηi , up to ±, which satisfies 0 = ηi ∧ dηi , by the Frobenius theorem. Denote 3-dimensional Euclidean space by X. Let M be the set of all orthonormal bases of the tangent spaces of X, with obvious bundle map x : M → X, so that each point of M has the form m = (x, e1 , e2 , e3 ) for some x ∈ X and orthonormal basis e1 , e2 , e3 of Tx X. The soldering 1-forms ω1 , ω2 , ω3 on M are defined by (v ωi )ei = x∗ v. Note: they are 1-forms on M , not on X. Let   ω1 ω ..= ω2  . ω3 As explained in appendix D, there is a unique matrix-valued 1-form π, valued in antisymmetric 3 × 3 matrices and known as the Levi-Civita connection, so that dω = −π ∧ ω, i.e.       ω1 0 −π3 π2 ω1 0 −π1  ∧ ω2  . d ω2  = −  π3 −π2 π1 0 ω3 ω3 Our triply orthogonal web is precisely a section X → M of the bundle map M → X on which 0 = ωi ∧ dωi for all i, hence an integral 3-manifold of the exterior differential system I on M generated by the closed 3-forms ω1 ∧ dω1 ,

ω2 ∧ dω2 ,

ω3 ∧ dω3 .

Using the equations above, I is also generated by π3 ∧ ω12 ,

π1 ∧ ω23 ,

π2 ∧ ω31 .

The integral manifolds coframed by ω1 , ω2 , ω3 are locally precisely the triply orthogonal webs. The reader can find the characters: s1 , s2 , s3 = 0, 3, 0. The integral elements coframed by ω1 , ω2 , ω3 form a manifold of dimension 12 (parameterized by choice of a point of M and 6 coefficients to determine values of π1 , π2 , π3 at each point of M ): involution. The reader familiar with Riemannian geometry will note that this proof works just as well for any 3-dimensional Riemannian manifold. For more on orthogonal webs in Euclidean space, see [7, 9, 35, 39]. 1.22∗ A question of Alin Albu-Schäffer: suppose we are given an analytic foliation of an open subset of 3-dimensional Euclidean space by surfaces. Is there, at least locally, a triply orthogonal web which has this as one of its three foliations?

10

Exterior differential systems

Generality of integral manifolds This section can be omitted without loss of continuity.

If I describe some family of submanifolds as integral manifolds of an exterior differential system, you might find a different description of the same submanifolds as integral manifolds of a different exterior differential system, perhaps on a different manifold, with different characters. A function f (x) of one variable is equivalent information to having a constant f (0) and a function df /dx of one variable. We will solve ∇×u = f −u for an unknown vector field u in 3-dimensional Euclidean space two ways: an elementary approach on page 83, and following Cartan’s strategy in problem 2.5 on page 21. The first approach uses 1 function of 1 variable and 2 functions of 2 variables. The second approach uses 3 constants, 3 functions of 1 variable and 2 functions of 2 variables. Lagrangian submanifolds of Cn depend on 1 function of n variables: those which are graphs y = y(x) are of the form y=

∂S ∂x

for some potential function S(x), unique up to adding a constant. On the other hand, the proof of the Cartan–Kähler theorem builds up each Lagrangian manifold from a choice of 1 function of 1 variable, 1 function of 2 variables, and so on. We will see that Cartan’s proof of the Cartan–Kähler theorem constructs the general solution of any involutive exterior differential system, using s0 constants, s1 functions of 1 variable, s2 functions of 2 variables, and so on. We “trust” the last nonzero sp of an involutive exterior differential system to give the generality of integral manifolds: they depend on sp functions of p variables, but we don’t “trust” s0 , s1 , . . . , sp−1 . Immersed plane curves are the integral curves of I = 0 on M = R2 . Any integral flag has s0 , s1 = 0, 1. Immersed plane curves are also the integral curves of the ideal J generated by ϑ ..= sin φ dx − cos φ dy on M ..= R2x,y × Sφ1 . Here s0 , s1 = 1, 1.

What could go wrong?

What could go wrong? On M ..= R3x,y,z , take the ideals I generated by dx ∧ dz, dy ∧ dz and J generated by dx ∧ dz, dy ∧ (dz − y dx). At the origin, these differential forms are identical, so the integral elements and characters are the same at that point. But I has integral surfaces z = constant, while J does not have integral surfaces. 1.23 Compute characters and dimensions of spaces of integral elements for these. Show that the Cartan–Kähler theorem does not apply. Find their integral surfaces.

Generalizations The Cartan–Kähler theorem also holds for holomorphic exterior differential systems on complex manifolds, formal power series exterior differential systems, and Denjoy–Carleman exterior differential systems, with the same proof. In particular, any involutive smooth exterior differential system is also a formal power series system about each point, so has formal power series solutions, perhaps divergent. The theorem also holds for certain smooth systems [21, 22, 38].

11

Chapter 2

Tableaux

A tableau is a matrix encoding an exterior differential system, a computational tool to organize the linear algebra needed to uncover the characters.

Recall triply orthogonal webs on page 8: an exterior differential system generated by 1-forms θa and by 3-forms of the form ω 12 ∧ π 3 , ω 31 ∧ π 2 , ω 23 ∧ π 1 , with coframe θa , ω i , π α . Write these 3-forms as rows of a matrix wedge product:  3   12  π 0 0 ω  0 π 2 0  ∧ ω 13  0 0 π1 ω 23

Definition Take an exterior differential system I on a manifold M . We are looking for p-dimensional integral manifolds. Take a point m ∈ M . Let ∗ Im ..= { ϑm ∈ Λ∗ Tm M | ϑ ∈ I }. 1 We carry out all of our work below modulo the ideal (Im ) ⊆ Im . ∗ 1 a. Take a basis ω i , π α of Tm M/Im . 1 b. Write a set of forms generating Im /(Im ) in a column

 1 ϑ ϑ2    ϑ =  . .  ..  ϑr 13

14

Tableaux

c. Let ω ij ..= ω i ∧ ω j , and so on. Let             ω=            

ω1 ω2 ω 12 ω3 ω 13 ω 23 ω 123 .. . ωp .. . ω 1···p

            ,           

a column vector of all wedge products of the ω i . Arrange by grades: grade j consists of all forms ω ···j which are wedge products of 1-forms from among ω 1 , . . . , ω j , and must contain ω j . We sometimes mark grades with horizontal lines. (We can drop any entry of ω if it doesn’t appear in the forms ϑa in our generating set, expanded in our basis.) d. Write out ϑ = \$ ∧ ω + . . . for a matrix \$, the tableau, so that each matrix entry is a linear combination of π α , and the . . . consists of 1. terms with only ω in them, no π α 1-form wedged into them, the torsion and 2. terms with two or more π α 1-forms wedged into them, the nonlinearity. We sometimes draw vertical lines in \$, marking out grades at widths matching the grade heights in ω. The nonlinearity we assign grade p. e. A polar is an entry of \$ linearly independent of all entries found in all earlier grades and above or to the left in the same grade. Highlight all polars. In practice, each polar is often one of the π α of our basis; we can always change basis to arrange this. 1 ; declare the basis elements to be polars of grade f. Take any basis θa of Im zero.

The character sj of the tableau is the number of polars in grade j.

15

Definition

Suppose I is an exterior differential system spanned by 1-forms θ1 , θ2 , θ3 , a 2-form θ1 ∧ ω 3 + ω 1 ∧ π 1 + ω 2 ∧ π 2 + ω 3 ∧ π 3 and a 3-form π 123 − ω 12 ∧ π 3 + ω 13 ∧ π 2 − ω 23 ∧ π 1 , for a coframing

θ1 , θ2 , θ3 , ω1 , ω2 , ω3 , π1 , π2 , π3 .

Dropping multiples of the θa : no 1-forms a 2-form ω 1 ∧ π 1 + ω 2 ∧ π 2 + ω 3 ∧ π 3 and a 3-form − ω 12 ∧ π 3 + ω 13 ∧ π 2 − ω 23 ∧ π 1 + . . . . So modulo θ1 , θ2 , θ3 :     0 ω1 ∧ π1 + ω2 ∧ π2 + ω3 ∧ π3 ϑ= + , π 123 −ω 12 ∧ π 3 + ω 13 ∧ π 2 − ω 23 ∧ π 1 

=−

π1

π2

0

0

0 π

3

π3 0

0 −π

0 2

π1

s1

s2

s3

1

2

0

!

    ∧    

ω1 ω2 ω 12 ω3 ω 13 ω 23

       0 + .  π 123   

Recall that Lagrangian submanifolds are integral manifolds of ϑ ..= dx1 ∧ dy 1 + dx2 ∧ dy 2 + · · · + dxn ∧ dy n ,  1 ω 2    ω  = − π1 π2 . . . πn ∧  .   ..  s1 s2 . . . sn ωn 1 1 ... 1 with ω i ..= dxi , π i ..= dy i .

16

Tableaux

On M = R5x,y,u,ux ,uy , let I be generated by θ ..= du − ux dx − uy dy, ϑ ..= dux ∧ dy − duy ∧ dx, and note

dθ = −dux ∧ dx − duy ∧ dy

also belongs to I. An integral surface X on which 0 6= dx ∧ dy is locally the graph of a harmonic function u = u(x, y) and its derivatives ux = ∂u ∂x , uy = ∂u . Taking ∂y ω 1 ..= dx, ω 2 ..= dy, π 1 ..= dux , π 2 ..= −duy , gives tableau 

dθ ϑ



=−

π1

−π 2

π2

π1

!

 ω1 ∧ . ω2 

2.1 As in our example of triply orthogonal webs, construct an exterior differential system whose integral 4-manifolds are foliations of open subsets of 3-dimensional Euclidean space. Write out the tableau and find the characters.

Torsion The tableau is adapted to the flag Ej ..= (0 = θa = π α = ω j+1 = · · · = ω p ). Any flag has many tableau adapted to it. 2.2 What are the polar equations of each Ej ? Prove that the characters as defined above are the characters as defined in chapter 1, and the polars are a basis of the polar equations. 2.3 Prove that torsion vanishes just when the flag is integral. Take an exterior differential system generated by 1-forms θ1 , θ2 with    1  1   12  1 π 0  θ ω ω  mod θ1 , θ2 . d 2 =− ∧ 2 + 0 2 3 θ ω π π Note that dθ1 , dθ2 generate I/(I 1 ). The term ω 12 is the torsion, as it has no π in it.

17

Borrowing

Let π 01 ..= π 1 + ω 2 :   1 π 01 θ d 2 = − θ π2

  1 0  ω ∧ ω2 π3

mod θ1 , θ2 ;

we absorb the torsion. Changing bases to arrange that torsion vanishes is absorbing the torsion A tableau can only examine integral elements coframed by the ω i . Torsion absorbs just when there is such an integral element. The exterior differential system I is the true fundamental geometric object; the choice of tableau is like a choice of coordinate system: a magnifying glass with which to examine I.

Borrowing Recall triply orthogonal webs on page 8 had exterior differential system generated by 1-forms θa and by 3-forms of the form ω 12 ∧ π 3 , ω 31 ∧ π 2 , ω 23 ∧ π 1 , with coframe θa , ω i , π α :  π3 0 0   0 π2 0  0 0 π1 s2

s3

1

2

 ω 12   13  ∧  ω   ω 23

Warning: these s1 , s2 , s3 are not the characters we computed in chapter 1.

Reconsider the same example, with new choices of 1-forms ω i . Let ω 01 ..= ω 1 , ω 02 ..= ω 2 , ω 03 ..= ω 1 − ω 2 + ω 3 . Write this as  1   ω ω 01 ω 2  =  . ω 02 3 01 02 03 ω −ω + ω + ω

18

Tableaux

In these 1-forms, the tableau is:  π3 0 0   π2 0 0  π1 0 0 s2

s3

3

0

 ω 012   013  ∧  ω   023 ω

These are the characters we computed in chapter 1. We have borrowed polars from later grades into earlier. We change the choice of integral flag from E1 = (0 = ω 2 = ω 3 = θa = π α ), E2 = (0 = ω 3 = θa = π α ), E3 = (0 = θa = π α ), to E10 = (0 = ω 02 = ω 03 = θa = π α ) E20 = (0 = ω 03 = θa = π α ), E3 = (0 = θa = π α ) unchanged.

2.4 Prove that the characters depend only on the flag. Borrowing as many polars as we can, and perhaps permuting the coframe, the integral flag has largest s1 among all integral flags at that point. Subject to that s1 , it has largest s2 , and so on. Take a row which represents a k-form. Permuting the ω i , we can get any polar in that row to appear in grade k − 1: π α ∧ ω 1... k−1 . If there is another polar in that row, say in grade `, add a multiple of ω k to ω ` to borrow it to grade k. Continue in this way: for one particular row, representing a k-form in I, we arrange polars in successive grades, starting at grade k − 1, all followed by any nonpolar entries in that row. Write wedge products ω i1 ...iq with i1 < · · · < iq . Order any two wedge products by last entry iq , then by next to last, and so on. Borrow to have polars arising in sorted order before any other entries. Since this occurs for some linear transformation of ω i , it also occurs for all linear transformations of ω i except for those with certain minors vanishing. We can thus borrow simultaneously for all rows, by generic linear transformation.

19

Borrowing

π1

0

0

π1

ω 12 ω3

! ∧

!

has a polar appearing in grade 2. Permuting indices 1 and 3: ! ! ω1 0 π1 ∧ ω 23 π1 0 puts it in grade 1. The torsion is absorbable just when there is an integral element E = (π = pω). Absorb the torsion by subtracting pω from π. Thus there is a torsion free tableau just when there is a generic torsion free tableau. Take a tableau π1

0

!

π2 π3

ω1 ∧ ω2 



+

π4 ∧ π2 0

!

with a polar in the nonlinearity. Add ω 2 to π 2 to get the polar to appear in the tableau: !  1    4  0 ω π ∧ π 02 π1 π4 ∧ − + . ω 12 0 ω2 π 02 π 3 This produces torsion, but we can absorb it.



π1

π2



 1 ω 3 + π ∧ π2 ω2

has a polar in the nonlinearity, but we absorb it by ω 02 ..= ω 2 − π 3 .

Some tableaux have nonabsorbable polars in the nonlinearity:    1 ω 3 4 + π ∧ π . π1 π2 ∧ ω2 Any polar in the nonlinearity can be demoted to a new ω i 1-form, but we won’t need to do so.

20

Tableaux

Integral elements Picture a tableau       

π1

π4

2

5

π

π

π3 π − π2 1

π6

  1 ω  π −π   ∧ ω 2  .  ω3 π1 + π2  3 π 1

π2 0

5

Take any 3-dimensional linear subspace E of a tangent space of M coframed by ω 1 , ω 2 , ω 3 and on which θa = 0. Then E has coefficients π 1 = p11 ω 1 + p12 ω 2 + p13 ω 3 and so on. Plug in to the tableau to find equations for integral elements. Since ω ij = −ω ji , each tableau entry in column i has coefficient of ω j exactly equal to the tableau entry in column j coefficient of ω i : p12 = p41 ,

p13 = p61 ,

p43 = p62 ,

p22 = p51 ,

p23 = p11 − p51 ,

p53 = p12 − p52 ,

p32 = p21 ,

p33 = p11 + p21 ,

p23 = p12 + p22 ,

p12 − p22 = 0,

p13 − p23 = p31 ,

0 = p32 .

Again imagine an exterior differential system generated by 1-forms θa and by 3-forms of the form ω 12 ∧ π 1 , ω 31 ∧ π 2 , ω 23 ∧ π 3 , with coframe θa , ω i , π α . (E.g. triply orthogonal tableau:    ω 12 0 0 π1    13  0 π2 0  ∧  ω   ω 23 3 π 0 0 s2

s3

1

2

3-dimensional integral elements:  1  0 π π 2  = p21 π3 p31

p12 0 p32

webs; see page 8.) The   

  1 ω p13 p23  ω 2  0 ω3

a 6-dimensional space of integral elements at each point. s1 + 2s2 + 3s3 = 0 + 2(1) + 3(2) = 8 > 6,

21

Example: Lie’s third theorem

involution fails. The same example, but borrow:  1  ω ω 2  = 

 ω 01 , ω 02 ω 01 + ω 02 + ω 03

ω3 yielding tableau: 

−π 1

0

π3

  −π 2  π3

π2

0

0

0

s2

s3

3

0

 ω 012   013  ∧  ω   ω 023

s1 + 2s2 + 3s3 = 0 + 2(3) + 3(0) = 6, involution: there are 3-dimensional integral manifolds.

2.5 Write the equation ∇ × u = f − u, which we unravel in detail on page 83, as an exterior differential system. Find a tableau. Can you absorb torsion? What submanifold contains the integral manifolds? Is the exterior differential system in involution on that submanifold?

Example: Lie’s third theorem This section can be omitted without loss of continuity.

In this section we assume familiarity with Lie groups [33]. Lie’s third theorem: every Lie algebra g, say of dimension p, is isomorphic to a Lie algebra of vector fields spanning every tangent space on a p-dimensional manifold. This theorem is a first step in constructing a Lie group with a given Lie algebra. Since we employ differential forms, it is easier for us to prove the dual statement about the dual 1-forms to those vector fields. A Maurer–Cartan form is a 1-form ξ valued in a Lie algebra g, defined on an open subset U ⊂ Rp , so that, at every point x ∈ U , ξx : Rp → g is a linear isomorphism and dξ + 21 [ξξ] = 0. 2.6 Explain how a Maurer–Cartan form determines such vector fields and vice versa. Theorem 2.1 (Lie III). Any Lie algebra has a Maurer–Cartan form.

22

Tableaux

Proof. Choose a basis e1 , . . . , ep ∈ g and write out the Lie bracket in the basis: [ei ej ] = ckij ek . Any such ξ will then be ξ = ξ i ei , for a coframing ξ i = gji (x)dxj , with gji (x) an invertible matrix for each x ∈ U . We want these ξ i to satisfy j k p i i .. 0 = dξ i + 12 cjk i ξ ∧ ξ . Let M = R × GL(p, R) with coordinates x , gj . Take i the exterior differential system I generated by the components ϑ of the 2-form ϑ = d(g dx) +

1 [g dx, g dx] . 2

2.7 Use the Jacobi identity, either in components or directly, to see that 0 = dϑ. Let ω i ..= dxi . i 2.8 If we want ϑi = πji ∧ ω j , then πji = dgji + qjk (c)dxk where q = q(c) is some quadratic polynomial expression in the coefficients ckij . Prove this by computing q(c).

 

1

ϑ   ϑ2       . =  ..      ϑp

π11 π21 . . . πp1 π12 π22 .. .. . . π1p π2p

 1 ω   2  . . . πp   ω 2  ∧  . ..   ...  ... .    ωp p . . . πp

s1 s2 . . . sp p

p ... p

Integral elements of dimension p: πji = pijk ω k with pijk = pikj , so the space of integral elements has dimension dim M + p(p + 1)/2: involution. 2.9 Uniqueness: prove that any two Maurer–Cartan forms for the same Lie algebra are locally identified by a diffeomorphism.

Example: surface invariants This section can be omitted without loss of continuity.

We suppose that the reader has read appendix D. Does every quadratic form on a plane through a point arise as the shape operator on the tangent plane of some surface? If we had such a surface, its adapted frame bundle would satisfy ω3 = 0, γ13 = a11 ω1 + a12 ω2 , γ23 = a21 ω1 + a22 ω2 ,

23

Example: surface invariants

with aij = aji . Let V be the set of all symmetric 2 × 2 matrices, with typical element written as   a11 a12 a= . a21 a22 On the manifold V × ⌜E3 , take the exterior differential system I generated by ω3 , γ13 − a11 ω1 − a12 ω2 , γ23 − a21 ω1 − a22 ω2 . Every surface in E3 has frame bundle an integral manifold. 2.10 Prove that any I-integral manifold coframed by ω1 , ω2 , γ12 is locally the frame bundle of a surface. Summing over i, j, k, ` = 1, 2, let Daij ..= daij − aik γkj − ajk γki ,    0 0 0 ω3  d γ13 − a11 ω1 − a12 ω2  =  Da11 Da12 0  γ23 − a21 ω1 − a22 ω2 Da12 Da22 0 s1

s2

s3

2

1

0

  ω   1  ∧ ω2 ,  γ12

Integral elements coframed by ω1 , ω2 , γ12 are Daij = aijk ωk + Aij γ12 . Plug into the tableau: integral elements have aijk symmetric in all 3 indices, and Aij = 0. Each integral element is identified with III ..= e3 aijk ωi ωj ωk , the third fundamental form on any integral manifold arising from a surface. The space of integral elements is 4-dimensional at each point of our manifold, involution: there is an integral manifold coframed by ω1 , ω2 , γ12 through every point of V × ⌜E3 . Theorem 2.2. Shape operators are arbitrary. To be precise, take a point of E3 , a plane through that point, a symmetric bilinear form and a symmetric trilinear form on that plane, valued in the perpendicular line. There is a surface in E3 through that point, tangent to that plane, and with that bilinear form as shape operator and that trilinear form as third fundamental form. 2.11 Prove that there are surfaces in 3-dimensional Euclidean space preserved by no rigid motions except the identity. 2.12 Are the shape operators of the leaves of a triply orthogonal web arbitrary?

Chapter 3

Example: almost complex structures

This chapter can be omitted without loss of continuity. As an application of the Cartan– Kähler theorem, we demonstrate the integrability of torsion free almost complex structures.

Example: the Cauchy–Riemann equations This section can be omitted without loss of continuity.

Take several complex variables z 1 , . . . , z n , with real and imaginary parts z µ = xµ + iy µ . A holomorphic function is a complex valued function f (z 1 , . . . , z n ) satisfying the Cauchy–Riemann equations [8, 15, 18, 32, 34] ∂f ∂f = i µ, µ ∂x ∂y

µ = 1, 2, . . . , n.

In real and imaginary parts f = u + iv, this becomes ∂u ∂v = , µ ∂x ∂y µ ∂v ∂u = − µ. ∂xµ ∂y Let M ..= Rx4n+2 on which we take the exterior differential system I µ ,y µ ,u,v,p ,q µ µ generated by θ1 ..= du − pµ dxµ − qµ dy µ , θ2 ..= dv + qµ dxµ − pµ dy µ . 

  1 dp1 dp2 θ d 2 =−− θ −dq1 −dq2

...

dpn

dq1

...

dqn

...

−dqn

dp1

...

dpn

s1

s2

...

sn

sn+1

...

s2n

2

2

...

2

0

...

0

25

 dx1  dx2        ..   .     ∧ dxn   1  dy     ..   .  dy n

26

Example: almost complex structures

So

dim M + s1 + 2s2 + · · · + 2ns2n = 4n + 2 + n(n + 1).

Integral elements at each point (x, y, u, v, p, q) are      ν dpµ r −sµν dx = µν dqµ sµν rµν dy ν with r, s symmetric in lower indices. So the space of integral elements has dimension 4n + 2 + n(n + 1), involution with general solution depending on 2 functions of n variables. For a complex notation, let ω µ ..= dxµ +i dy µ , θ ..= θ1 +iθ2 , πµ ..= dpµ +i dqµ and ω µ¯ ..= ω µ = dxµ − i dy µ . A tableau in complex differential forms will have terms expressed in wedge products with ω µ , ω µ¯ . But for the Cauchy–Riemann equations we find  ω1   ω2    πn ∧  .   ..  ωn 

dθ = −



π1

π2

...

with no ω µ¯ terms. The characters count complex polars, so double them to count real and imaginary parts: s0 , s1 , s2 , . . . , sn = 2, 2, . . . , 2. Integral elements are πµ = pµν ω ν , with complex numbers pµν = pνµ , so n(n + 1)/2 complex numbers, hence n(n + 1) real and imaginary parts.

Example: almost complex structures This section can be omitted without loss of continuity.

We can turn a real vector space into a complex vector space, just by picking a real linear map J so that J 2 = −I. An almost complex structure on an even dimensional manifold M is a choice of complex vector space structure on each tangent space, analytically varying, in that is described by an analytic map J : T M → T M , acting linearly on each tangent space, with J 2 = −I. Complex Euclidean space has its usual almost complex structure J(x, y) = (−y, x), preserved by biholomorphisms (i.e. holomorphic diffeomorphisms) between open sets, as their derivatives are complex linear maps. Any complex manifold M has an almost complex structure, defined by using holomorphic coordinates to identify locally with complex Euclidean space. On an even dimensional manifold, a complex coframing is a collection of complex valued 1-forms ω µ so that their real and imaginary parts coframe. It is complex linear for an almost complex structure if each ω µ is a complex linear √ map on each tangent space, i.e. ω µ ◦ J = −1ω µ for every µ. 3.1 Prove that every almost complex structure has, near each point, some complex linear complex coframing.

27

Example: almost complex structures

3.2 Prove that every complex coframing is complex linear for a unique almost complex structure. Complex coframings are a useful way to exhibit almost complex structures. The complex coframing ω 1 ..= dz, ω 2 ..= dw − w ¯ d¯ z yields a unique almost complex structure on the space parameterized by two complex variables z, w. 3.3 Prove that any two complex coframings ω µ , ω 0µ yield the same almost complex structure just when ω 0µ = gνµ ω ν for a unique matrix g = (gνµ ) of functions. Any complex differential form is expressed in any complex coframing ω µ as a sum of wedge products of ω µ and ω ¯ µ . Following convention, write ω ¯ µ as ω µ¯ . µ A (p, q)-form is a differential form expressed with p factors of ω and q of ω µ¯ . For example, ω µ is (1, 0) while ω µ¯ is (0, 1). In particular, dω µ = tµµσ ω ν ∧ ω σ + tµνσ¯ ω ν ∧ ω σ¯ + tµν¯σ ω ν¯ ∧ ω σ + tµν¯σ¯ ω ν¯ ∧ ω σ¯ , for unique complex valued functions t, antisymmetric in lower indices. The exterior derivative of any (p, q)-form is uniquely expressed as a sum of forms with (p, q) raised by (2, −1), (1, 0), (0, 1) or (−1, 2). We thus split up d = τ + ∂ + ∂¯ + τ¯ : τ ω µ = 0,

∂ω µ = tµνσ ω ν ∧ ω σ , ¯ µ = tµ ω ν ∧ ω σ¯ + tµ ω σ¯ ∧ ω ν . τ¯ω µ = tµν¯σ¯ ω ν¯ ∧ ω σ¯ , ∂ω νσ ¯ σ ¯ν Let ω ..= (ω µ ). Expanding out in the coframing, we find that 0 = τ = τ¯ on all differential forms if and only if τ¯ω = 0. Change coframing: replace ω by some gω. d(gω) = dg ∧ ω + gdω, expands out to τ (gω) = 0, τ¯(gω) = g τ¯ω,

∂(gω)= (∂g) ∧ ω + g ∂ω, ¯ ¯ ∧ ω + g ∂ω. ¯ ∂(gω)= (∂g)

Hence τ¯ω = 0 just when τ¯(gω) = 0. So vanishing of τ¯ is a property of the almost complex structure. 3.4 Prove that τ¯ = 0 just when τ = 0. 3.5 Construct a tensor whose vanishing is equivalent to τ¯ = 0.

28

Example: almost complex structures

The complex coframing ω 1 ..= dz, ω 2 ..= dw − |w|2 d¯ z has τ¯ω 1 = 0, τ¯ω 2 = −w dw ¯ ∧ d¯ z, ¯

¯

= w ω1 ∧ ω2 , 6= 0. Therefore the associated almost complex structure is not a complex structure. A complex valued function f : M → C on an almost complex manifold is holomorphic if df is complex linear. Lemma 3.1. An almost complex manifold which admits holomorphic functions with arbitrary complex linear differentials at each point has τ¯ = 0. Proof. Denote by 2n the dimension of M as a real manifold. Since this problem is local, we can assume that M has a global complex linear coframing ω µ . Take the manifold M 0 ..= M × Cz × CnZ , and the exterior differential system generated by dz−Zµ ω µ . Any integral manifold on which ω µ are complex-linearly independent is locally the graph of a holomorphic function, and vice versa. The tableau d(dz − Zω) = −dZ ∧ ω − Z dω, = −DZ ∧ ω − Z τ¯ω, where

DZµ ..= dZµ + Zσ (tσνµ ω ν + 2tσν¯µ ω ν¯ ).

The torsion, where Z 6= 0, consists of the expression Z τ¯ω = Zµ τ¯ω µ . 3.6 Find all holomorphic functions for the almost complex structure of the complex coframing ω 1 ..= dz, ω 2 ..= dw − w ¯ d¯ z Any wedge product π ∧ ω of complex valued 1-forms can be rewritten as a real wedge product in real and imaginary parts  1   1  π −π 2 ω −ω 2 ∧ . π2 π1 ω2 ω1

29

Example: almost complex structures

If π ∧ ω occurs in a tableau, at some grade, it contributes 2 linearly independent 1-forms:    1  2 π 1 −π 2   ∧ ω −ω . ω2 ω1 π2 π1 Count with a complex tableau as if it were real linear, but double the characters. Theorem 3.2. An analytic almost complex structure has τ¯ = 0 just when it arises from a complex structure. This theorem remains true with milder assumptions than analyticity [17]. Proof. On a complex manifold with holomorphic coordinates z µ , the 1-forms ω µ = dz µ are complex linear for the standard almost complex structure, and have dω µ = 0, so no torsion. Take an almost complex structure J which has τ¯ = 0. Our problem is to construct local holomorphic coordinate functions locally identifying J with the standard complex structure on complex Euclidean space. Again take the exterior differential system generated by θ ..= dz − Zω:  ω1 2   ω  ∧  . .  ..  ωn 

dθ = −



DZ1 DZ2

...

DZn

s1

s2

...

sn

2

2

...

2

Integral elements are DZµ = pµν ω ν , pµν = pνµ , n(n + 1)/2 complex constants, so n(n + 1) real constants, involution. So there are holomorphic functions with arbitrary differentials at a point, i.e. local holomorphic coordinates.

The complex coframing ω 1 = dz, ω 2 = dw − w d¯ z, determines a complex structure.

The expression τ¯ω µ = tµν¯σ¯ ω ν¯ ∧ ω σ¯ is antisymmetric in ν¯, σ ¯ . It vanishes if M has complex dimension 1, i.e. real dimension 2: every almost complex manifold of real dimension 2 is a Riemann surface.

30

Example: almost complex structures

In this example, we assume familiarity with matrix groups [33]. The manifold SU3 is the collection of all 3 × 3 complex unitary matrices z = (zµ¯ν ) of determinant 1. Write zµ¯ν to mean z¯µ¯ν , so that z ∗ has ∗ entries zµ¯ ¯µ . Unitarity is zµ¯ σ zν ¯σ = δµ¯ ν . Note that SU3 is a real ν = zν submanifold, not a complex submanifold, of the 3 × 3 complex matrices, as this unitarity equation is not complex analytic. Since SU3 is a submanifold of matrices, each tangent vector to SU3 is a matrix. The tangent space TI SU3 at the identity matrix, denoted su3 , is the set of all traceless skew adjoint 3 × 3 complex matrices. It is traditional to write the identity function on any group as g, so g(z) = z. The Maurer–Cartan 1-form ω ..= g −1 dg is a 1-form on SU3 , valued in su3 , i.e. to each tangent vector v ∈ Tz SU3 , which we identify with a matrix A, v ω = z −1 A. Write out ω as a matrix of complex valued 1-forms   ω1¯1 ω1¯2 ω1¯3 ω = ω2¯1 ω2¯2 ω2¯3  ω3¯1 ω3¯2 ω3¯3 with ωµ¯ν = −ων¯µ . 3.7 Prove that ω is invariant under left translation. 3.8 Calculate that dω = −ω ∧ ω. In matrix entries, dωµ¯ν = −ωµ¯σ ∧ ωσν¯ . Consider the coframing ω1¯1 + iω2¯2 , ω1¯2 , ω1¯3 , ω2¯3 , 3.9 Take exterior derivatives and find torsion vanishing: SU3 has a left invariant complex structure. 3.10 On the other hand, if we conjugate one of the last three 1-forms in the coframing, prove that we arrive at a left invariant almost complex structure which is not complex. 3.11∗ Prove theorem 3.2 on the preceding page by complexifying variables locally, and applying the Frobenius theorem.

Almost complex submanifolds This section can be omitted without loss of continuity.

An almost complex submanifold of an almost complex manifold M is a submanifold whose tangent planes are complex linear subspaces. Suppose that M has real dimension 2(p + q). Let’s look for almost complex submanifolds of dimension 2p. Take a complex linear coframing ω 1 , . . . , ω p , π 1 , . . . , π q . Take an almost complex submanifold of real dimension 2p on which ω 1 , . . . , ω p have linearly

Almost complex submanifolds

independent real and imaginary parts. Then on that submanifold, π = pω for a complex matrix p. Our almost complex submanifold is an integral manifold of the exterior differential system π = pω on M × Cpq . Let θ ..= π − pω. On any integral manifold 0 = τ¯θ = τ¯π − p τ¯ω. 3.12 Prove that, for any p > 1, if there is an almost complex submanifold of real dimension 2p tangent to any complex linear subspace of complex dimension p in any tangent space, then M is a complex manifold. A holomorphic curve, often called a pseudoholomorphic curve, in an almost complex manifold M is a map C → M from a Riemann surface, with complex linear differential. 3.13 Prove the existence of embedded holomorphic disks, i.e. embedded holomorphic curves C → M where C is the unit disk in the complex plane, in every analytic almost complex manifold, tangent to every complex line in every tangent space.

31

Chapter 4

Prolongation

In our next example, we will see what to do when the Cartan–Kähler theorem does not apply to an exterior differential system.

Notation In this chapter, for convenience of notation, we drop our convention of writing ω 12 to mean ω 1 ∧ ω 2 etc. We use freely the notation and terminology of appendix D.

Example: isometric immersion, involution fails Take a surface S with a Riemannian metric. Naturally we are curious if there is an isometric immersion φ : S → E3 , i.e. a map preserving the lengths of all curves on S. This surface (viewed from various angles)

is the image of an isometric immersion of a piece of this paraboloid

On the frame bundle ⌜S of oriented orthonormal frames, denote the soldering forms as ω = ω1 + iω2 and the connection by 1-form) α so that dω = iα ∧ ω 33

34

Prolongation

and dα = (i/2)Kω ∧ ω ¯ . On ⌜E3 there is a soldering 1-form ωi0 and a connection 0 0 0 0 0 0 1-form γij so that dωi0 = −γij ∧ ωj0 and dγij = −γik ∧ γkj . Let α0 ..= γ12 . 3 Suppose that there is an isometric immersion φ : S → E . Its adapted frame bundle X ..= Xφ ⊂ M ..= ⌜S × ⌜E3 is the set of all tuples (x, e1 , e2 , x0 , e01 , e02 , e03 ) where x ∈ S with orthonormal frame e1 , e2 ∈ Tx S and x0 ∈ E3 with orthonormal frames e01 , e02 , e03 ∈ Tx0 E3 , so that φ∗ e1 = e01 and φ∗ e2 = e02 . Let I be the exterior differential system on M generated by the 1-forms    0  ω1 − ω1 θ1 θ2  ..= ω20 − ω2  . θ3 ω30 Along X, all of these 1-forms vanish, while the 1-forms ω1 , ω2 , α coframe. 4.1 Prove that all integral manifolds coframed by ω1 , ω2 , α are locally frame bundles of isometric immersions. For the moment, we concentrate on asking whether we can apply the Cartan– Kähler theorem.       0 α0 − α 0 θ1 ω    1  ∧ ω2 d θ2  = − − (α0 − α) mod θ1 , θ2 , θ3 0 0   θ3 α 0 0 −γ13 −γ23 0 s1

s2

s3

2

1

0

Each 3-dimensional integral element has ω 0 = ω, so is determined by the linear equations giving γ10 , γ20 , γ30 in terms of ω1 , ω2 , α on which dθ = 0:  0     γ13 a b 0 ω1 0  γ23  =  b c 0 ω2  . 0 0 0 α α0 − α Therefore there is a 3-dimensional space of integral elements at each point. But s1 + 2s2 = 4 > 3: not involutive, so we can’t apply the Cartan–Kähler theorem. 4.2 Is the existence of a p-dimensional involutive integral element precisely the condition that p + s0 + s1 + · · · + sp ≤ dim M on every integral element of dimension p? There is another way to look at the failure of involution. In chapter 7, we will see that the Cartan–Kähler theorem can only apply if the generic integral line sits in an integral plane. The equations on integral lines are ω10 = ω1 , ω20 = ω2 , ω30 = 0. On any integral plane α0 = α. The generic integral line does not sit in an integral plane, because it doesn’t have to satisfy α0 = α.

35

What to do?

What to do? On every integral element, we said that  0   γ13 a b 0  γ23  = b c 0 0 α0 − α

  0 ω1 0 ω2  . 0 α

Make a new manifold M 0 ..= M × R3a,b,c , and differential system generated by    0   γ13 θ4 a b 0 θ5  ..=  γ23  − b c θ6 0 0 α0 − α

on M 0 let I 0 be the exterior   0 ω1 0 ω2  . 0 α

Prolongation What should we do if there are no involutive integral elements? Take an exterior differential system I on a manifold M . Denote by Grp M the Grassmann bundle of p-dimensional linear subspaces of tangent spaces of M . Take the subset M 0 ⊂ Grp M of all p-dimensional integral elements. Cut out any point of M 0 near which M 0 is not a submanifold of Grp M . Write out a tableau in 1-forms θ, ω, π. Write each linear subspace coframed by ω as the solutions of the linear equations π = pω, θ = qω for some constants p, q. On an open subset of Grp M , p, q are functions valued in some vector spaces. Pull back the 1-forms θ, ω, π to that open subset via the map (m, E) ∈ Grp M 7→ m ∈ M . The subset M 0 is cut out by the equation q = 0 and various equations among the p. On M 0 , let θ0 ..= π − pω The exterior differential system I 0 on M 0 generated by ϑ0 is the prolongation of I. More abstractly, without choosing a local basis of 1-forms: a 1-form θ on Grp M is contact if, at each point E ∈ Grp M , θE is the pullback of a cotangent vector vanishing on E. The ideal I 0 is generated by the restrictions to M 0 of the contact 1-forms. Any integral manifold X ⊂ M determines an integral manifold x ∈ X 7→ Tx X ∈ M 0 . Recall that the fibers of a map are the preimages of points. All integral manifolds on M 0 nowhere tangent to the fibers of M 0 → M arise in this way. Roughly speaking, prolongation is differentiation: 4.3 Write an exterior differential system associated to the partial differential equation ux = uyy . Consider the system of differential equations obtained by differentiating both sides of the equation once in each of the variable: ux = uyy (... )x

uxx = uyyx

(... )y

uxy = uyyy

Explain why the prolongation is associated to that system.

36

Prolongation

Some theorems prove that, under some hypotheses, exterior differential systems become involutive after sufficiently many prolongations [3] p. 255 theorem 3.2, [26] p. 68 theorem 4.11.

Example: isometric immersion, prolonging Returning to our example of isometric immersion of surfaces, prolong:    0     γ13 a b 0 ω1 θ4 0 θ5  ..=  γ23  −  b c 0 ω2  . θ6 0 0 0 α α0 − α Note that 0 = dθ1 , dθ2 , dθ3 modulo θ4 , θ5 , θ6 , so we can forget about them. Calculate the exterior derivatives:         Da Db 0  ω1 0 θ4       mod θ1 , . . . , θ6 0 d θ5  = −  Dc 0  ∧ ω2 +  Db θ6 α tω1 ∧ ω2 0 0 0 where

    da + 2bα + a1 ω1 + a2 ω2 , Da  Db  ..= db + (a − c)α + b1 ω1 + b2 ω2 , Dc dc + 2bα + c1 ω1 + c2 ω2 ,

where a1 , a2 , b1 , b2 , c1 , c2 can be chosen as we like, as long as a2 = b1 , b2 = c1 . (It is convenient to leave the freedom to choose these later.) The torsion is t ..= ac − b2 − K. This torsion clearly has to vanish on any 3-dimensional I 0 -integral element, i.e. every 3-dimensional I 0 -integral element lives over the subset of M 0 on which K = ac − b2 . To ensure that this subset is a submanifold, we let M00 ⊂ M 0 be the set of points where this equation is satisfied and at least one of a, b, c is not zero. Clearly M00 ⊂ M 0 is a submanifold, on which we find Da, Db, Dc linearly dependent. On M00 :       Da Db 0 θ4 ω1   d θ5  = − Db Dc 0  ∧ ω2  mod θ1 , . . . , θ6 .   θ6 α 0 0 0 s1

s2

s3

2

0

0

Example: isometric immersion, prolonging

There are 2 dimensions of integral elements at each point, involution: there is an integral manifold through each point of M00 , and in particular above every point of the surface. The prolongation exposes the hidden necessary condition for existence of a solution: the relation K = ac − b2 between the curvature of the surface and the shape operator. 4.4 Prove that any integral manifold of the exterior differential system constructed above coframed by ω1 , ω2 , α is locally the frame bundle of a isometric immersion. We can easily generalize these computations: Theorem 4.1. Take any surface S with Riemannian metric and a point x0 ∈ S. Denote the Gauss curvature by K. Take any 3-dimensional manifold X 0 with Riemannian metric, a point x00 , and a linear isometric injection F : Tx0 S → Tx00 X 0 . Let R0 be the sectional curvature tensor of X 0 on the image of F . Pick a nonzero quadratic form q on the tangent plane Tx0 S so that det q = K − R0 . Then there is an isometric immersion f of some neighborhood of x0 to X 0 , so that f 0 (x0 ) = F and so that f induces shape operator q at x0 . Write dK = K1 ω1 + K2 ω2 . It is convenient to pick our a1 , a2 , b1 , b2 , c1 , c2 to satisfy not just a2 = b1 , b2 = c1 , as required above, but also to satisfy K1 = 2bb1 − ac1 − ca1 , K2 = 2bb2 − ac2 − ca2 , so that 0 = c Da + a Dc − 2b Db, a simple relation among the differential forms in the exterior differential system. We can always do this near any point of M00 , by choice of a1 , a2 , b1 , b2 , c1 , c2 , using the fact that one of a, b, c is not zero. 4.5 Prove that every sufficiently small spherical cap on the unit sphere admits an isometric embedding to E3 not contained in a surface of revolution. For more on isometric immersions, see [16].

37

Chapter 5

Cartan’s test

We explain how involutivity can be seen as having the generic integral line lie in an integral plane and so on.

Generic points and generic submanifolds When we say that the generic point of some manifold has some property, we mean that those points which fail to have that property lie in a closed, nowhere dense set. The generic submanifold has some property if the property holds for all submanifolds whose tangent spaces avoid some closed, nowhere dense subset of the Grassmann bundle. The definition of generic in analysis is usually more sophisticated; we could use a more sophisticated definition without changing any of our proofs or results, but we won’t need to.

Cartan’s strategy Draw a point, an integral curve through that point, an integral surface through that curve, and so on up to some required dimension. We fail unless the integral curve is extendable, i.e. lies inside an integral surface, and so on. Integral curves are not always extendable. Don’t try to draw all integral curves, surfaces, and so on, but only the generic ones: ask whether the generic integral curve is extendable, and the generic integral surface, and so on.

Cartan’s test An integral line is extendable if it lies in an integral plane, and so on. We can imagine that extendability of generic integral manifolds is related to extendability of generic integral elements. 5.1 Prove that any integral element is extendable just when its codimension, in the space of integral elements, exceeds the rank of its polar equations. A 0-dimensional integral element is regular if its polar equations have locally maximal rank. Locally maximal rank is the same as locally constant rank, as linearly independent polar equations remain linearly independent nearby. An integral element is ordinary if it contains a regular hyperplane, and regular if in addition its polar equations have locally maximal rank. Regularity and ordinarity are difficult to test directly, so we will prove: 39

40

Cartan’s test

Theorem 5.1 (Cartan’s test). An integral element is involutive if and only if it is ordinary. Corollary 5.2. In each component of the space of integral elements, either none are involutive or the generic one is involutive. Ordinarity occurs precisely when the generic integral line sits in an integral plane, and so on: 5.2 Prove that an integral plane E2 is ordinary just when, for any integral line E1 ⊂ E2 , every integral line close enough to E1 sits in an integral plane.

Differentiating on the Grassmann bundle Each differential form ϑ yields an equation 0 = ϑ, satisfied by the integral elements of the exterior differential system it generates. At any given point m0 ∈ M , integral elements are points of the Grassmannian Grp Tm0 M . 5.3 Identify the polar equations at an integral element with the differentials of those equations on the Grassmannian. Corollary 5.3. Take an involutive integral element E of any exterior differential system I. Take a subsystem I 0 ⊆ I and a flag in E. If the I 0 -characters of that flag are the I-characters of E, then I and I 0 have the same integral elements near E, and so the same integral manifolds with tangent spaces near E. Proof. The I 0 -integral elements lie in a submanifold of the Grassmann bundle, with codimension equal to the number of linearly independent polar equations, by problem 5.3. This submanifold has the same dimension as the set of Iintegral elements, which it contains.

Extending integral elements Lemma 5.4. Every ordinary integral element is involutive. Proof. Impose p−1 generic linear constraints; they cut down our given ordinary integral element E to an integral line. They also cut down nearby integral elements to nearby integral lines. On the other hand, a line satisfying those generic constraints is integral just when it satisfies s0 linearly independent polar equations. Pick another p − 1 generic linear constraints. A line satisfying these will sum to our integral line to span an integral plane, just when it satisfies the s0 + s1 linearly independent polar equations of our integral line. By induction, the general p-dimensional integral element near E is cut out by solving ps0 +(p−1)s1 +· · ·+sp−1 equations. Solutions form an analytic manifold by problem 1.12 on page 6, of dimension dim M + s1 + 2s2 + · · · + psp . 5.4 Prove theorem 5.1.

41

View from a tableau

5.5 Prove that the generic linear subspace Ek ⊂ Ep of dimension k of an involutive integral element is involutive with the same characters s1 , s2 , . . . , sk as Ep , up to dimension k. 5.6 Take a subset E of a Grassmann bundle of a manifold. Let E⊥ be the set of all differential forms vanishing on all elements of E. Use the Cartan–Kähler to prove that E is an open subset of the set of involutive integral elements of an exterior differential system if and only if E⊥ is the unique maximal such system.

View from a tableau 5.7 Prove that, for any involutive integral element E ⊂ Tm0 M of an exterior differential system I, there is a tableau defined near m0 , with characters equal to those of I, generating an exterior differential system with the same characters and integral elements as I near E. 5.8 Prove in addition that locally (not just at a point) some change of coframing absorbs torsion. Put all polars π α of grade i into the entries of a column vector π i , with si rows. Write the equations for integral elements by plugging π i = pij ω j into the tableau. The grade of pij is i − j. For the moment, suppose there is no nonlinearity. Plug into the tableau to get linear equations solving for all coefficients pij for i < j in terms of various coefficients pβi , with a smaller subscript. Inductively, we solve for all negative grade coefficients in terms of semipositive grade coefficients (where semipositive means not negative). (There may be other equations as well, from entries that are not polars; ignore these.) We solve for p1i , for i = 2, 3, . . . , p, hence (p − 1)s1 equations. Similarly for the other grades, a total of (p − 1)s1 + (p − 2)s2 + · · · + sp−1 equations at least, on p(s1 + s2 + · · · + sp ) variables. Each integral element at this point is determined by the values of its semipositive coefficients. The number of semipositive coefficients is the difference between number of variables and number of equations: s1 + 2s2 + · · · + psp . We assumed no nonlinearity. Each nonlinearity term puts a quadratic or higher order expression into each of these equations. By the implicit function theorem, near an integral element, say arranged to be at the origin, higher order terms do not alter the possibility to solve the equations locally analytically. This tableau is only computed at one point. Extend the 1-forms in which it is expressed to be defined nearby as well. Careful: there might be differential forms in I which vanish at the point where we computed the tableau; we are

42

Cartan’s test

ignoring them. So when we extend the forms of the tableau to nearby points, we obtain a tableau for some subsystem of I, but maybe not all of I. The space of integral elements thus locally lies in a submanifold of the Grassmann bundle of dimension dim M + s1 + 2s2 + · · · + psp , parameterized by choices of point of M and semipositive grade coefficients. Near a given integral element, the space of integral elements is a submanifold of that dimension just when the tableau is involutive at that integral element. Involutivity holds just all other equations arising on integral elements are consequences of those above which solved for negative grade coefficient. The semipositive grade coefficients can then vary in some open set.

Differential equations All polars are linearly independent at our starting point, so we can assume that at some given point they are differentials of coordinate functions π α = duα , and that ω i = dxi . We can extend the ω i to anything we like at nearby points, so can assume ω i = dxi nearby. We can only arrange that the polars at nearby points are various multiples of dxi , duα . Careful: at nearby points, some other tableau entries might become linearly independent, i.e. more polars, and so more equations. Careful: there need not be any actual integral elements near this one; the equations we generate are only necessary conditions for an integral element. Nonetheless, by the implicit function theorem again, each equation of negative grade coefficients of each polar π α can be written as an equation of negative grade coefficients of some duα , and vice versa, hence as functions of the semipositive ones and the coordinates x, u. Let ui be the column vector of functions uα associated to polars π α of grade i. The equations become differential equations   ∂ui ∂uj = some function x, u, ≤j , ∂x>i ∂x for negative grade derivatives in terms of semipositive grade. We see one differential equation for each polar equation on each integral element in our flag. So far, we have not assumed involutivity: every integral manifold of every exterior differential system satisfies such equations. Involutivity is the condition that the tableau yields no more differential equations on integral manifolds. Choose any initial values ui (x1 , . . . , xi , 0, . . . , 0), so si functions of i variables, i = 0, 1, . . . , p. The differential equations become determined, so have local solutions near the origin by the Cauchy–Kovalevksaya theorem (theorem A.2 on page 80).

43

Compatibility and involutivity

The first differential equation solves for u0 along the x1 axis, from initial values at the origin. u

x1 x2 The second solves for u0 and u1 , in the x1 , x2 plane, from initial values along the x1 axis. u

x1 x2 But is the first differential equation still satisfied by u0 , at any constant value of x2 ? u

x1 x2 In other words, we need to see that these equations are compatible with one another. 5.9 Write out these differential equations for the exterior differential system of the Cauchy–Riemann equations in two complex variables. Can you solve these differential equations, by specifying initial data as above? 5.10 Prove that any involutive integral element of dimension p, whose characters all vanish except perhaps sp−1 and sp , is tangent to an integral manifold. 5.11 Use the Cartan–Kähler theorem to prove that every involutive integral element of any exterior differential system lies tangent to a leaf of a foliation by integral manifolds, foliating some open set. How many functions of how many variables do such foliations depend on? Prove that any involutive integral manifold is covered in open sets, each of which is a leaf in a foliation by integral manifolds, foliating some open set.

Compatibility and involutivity Geometrically, we sweep a point into an integral curve, sweep that into a surface, and so on. Compatibility asks that the surface is an integral surface, and so on. Suppose that the differential equations are compatible, i.e. we can pick any

44

Cartan’s test

initial values in the domain of our coordinates, and obtain an integral manifold. Then we can also pick initial values for the same differential equations nearby in those coordinates. The space of integral elements, parameterized by the semipositive derivatives ∂uj , ∂x≤j has the predicted dimension at all nearby points. So compatibility implies involutivity. An incompatibility, arising as an additional differential equation, might vanish, perhaps to high order, at the particular point where we are working, but obstruct at nearby points. So we check the dimension of the space of integral elements nearby. Incompatibilities can arise from commuting partial derivatives in our differential equations. Any exterior differential system is closed under exterior derivative, expressing the commutativity of first partial derivatives. So we expect that all incompatibilities are already present in the differential equations. So we expect that incompatibilities force the dimension of nearby integral elements below the predicted dimension. So we expect that compatibility is involutivity; chapter 7 gives a proof.

Generality of integral manifolds This section can be omitted without loss of continuity.

As above, Cartan’s strategy constructs integral manifolds of an involutive system by solving a sequence of determined equations, with initial data si functions of i variables, i = 0, 1, 2, . . . , p. Count Taylor coefficients of initial data: the Taylor series of order k of integral manifolds, at a chosen point in our coordinates, form a manifold of dimension       k k+1 k+p−1 s0 + s1 + s2 + · · · + sp . 0 1 p−1 As we vary k, these dimensions of determine all of the characters. Any other choice of determined systems of differential equations, giving rise to the same integral manifolds (or at least to the same Taylor series of integral manifolds at each order), injectively on each order of Taylor series, also has general solution depending on s0 constants, s1 functions of one variable, s2 functions of two variables, and so on.

Deformation This section can be omitted without loss of continuity.

A local deformation of an integral manifold X is an analytic map φ defined on an open subset of X × R containing X × { 0 } so that, for each constant t, x 7→ φ(x, t) is an integral manifold, where defined, with φ(x, 0) = x. The

45

Prolongation

velocity of the deformation is the projection of

∂φ ∂t

t=0

to the normal bundle

of X. By deforming initial data, any integral manifold of any determined system is covered in open sets admitting local deformation with velocity any solution of the linearization. We have not yet justified Cartan’s strategy to prove the Cartan–Kähler theorem, but we can already see that, since the strategy is a sequence of determined problems, we can apply the same reasoning; any involutive analytic integral manifold of any exterior differential system is covered in open sets admitting local deformation with velocity any solution of the linearization. Local analytic velocity fields exist in the same generality as integral manifolds, solving linear determined problems. The Holmgren uniqueness theorem [20] p. 80 proves that, on any analytic involutive integral manifold, any continuously differentiable velocity field is uniquely determined by its continuously differentiable initial data at each step in the sequence of linear determined problems. However, existence of continuously differentiable velocity fields is not guaranteed.

Prolongation This section can be omitted without loss of continuity.

When we prolong, we form new equations π = pj ω j for each polar. Only the semipositive coefficients pi≤i can vary freely. (For noninvolutive systems, not all of them vary freely.)  1   1   1 dp1 ∗ ∗ ... ∗ π − p1i ω 1 ω π 2 − p2i ω 1  dp21 dp22 ∗ . . . ∗  ω 2        d  = − . .. .. .. ..  ∧  ..  .. .    . . . . .   .  . p p p i p dp1 dp2 . . . . . . dppp ωp π − pi ω modulo torsion, since we haven’t included dπ i terms and dω i terms. The ∗ terms are dpij of negative grade, each solved for in terms of semipositive grade, so generate no polar. There is no nonlinearity. Some of these dp might be linearly dependent, due to noninvolutivity, so we don’t know where to highlight polars, but the polars lie in some spots in among these semipositive dp, on or below the diagonal. Consider the characters s0j of the prolongation. There are s1 + s2 + · · · + sp rows to this tableau, one for each polar, each representing a 1-form in I 0 . Grade zero in I 0 also includes I 1 , so s00 = s0 + s1 + · · · + sp . In the first column, there is at most one dp polar in each row: s01 ≤ s1 + s2 + · · · + sp . In the second column,

s02 ≤ s2 + s3 + · · · + sp ,

46

Cartan’s test

and so on, with finally s0p ≤ sp : the last nonzero character cannot increase. These inequalities are equalities just for involutive exterior differential systems.

Chapter 6

The characteristic variety

We give a geometric description of the characteristics of the associated partial differential equations of exterior differential systems.

Linearization The reader unfamiliar with linearization of partial differential equations, or characteristics, might look at appendix B. Take an exterior differential system I on a manifold M , and an integral manifold X. Suppose that the flow of a vector field v on M moves X through a family of integral manifolds. The tangent spaces of X are carried by the flow of v through integral elements of I. Equivalently, the flow pulls back each form in I to vanish on X. So 0 = Lv ϑ|X for any ϑ ∈ I. 6.1 Prove that all vector fields v tangent to X satisfy this equation. More generally, suppose that E ⊂ Tm M is an integral element of I. If a vector field v on M carries E through a family of integral elements, then 0 = Lv ϑ|E for each ϑ ∈ I. 6.2 Compute Lv ϑ|E in coordinates. Take any submanifold X. Suppose that a differential form ϑ vanishes on the linear subspace E ..= Tm X. Writing X as the graph of some functions, the expression ϑ|X , as a nonlinear first order differential operator on those functions, has linearization ϑ 7→ Lv ϑ|E . That linearization is applied to sections v of the normal bundle T M |X /T X. 6.3 In coordinates, prove that the linearized operator at the origin of our coordinates depends only on the integral element E = Tm X, not on the choice of submanifold X. Linearize any exterior differential system about any integral element by linearizing its differential forms, i.e. by linearizing the differential equations given by asking that those forms vanish on submanifolds. The linearization at p a p-dimensional integral element depends only on Im : the set of all values ϑm of forms in I p . 47

48

The characteristic variety

∗ 6.4 If P ..= E⊥ ⊂ Λ∗ Tm M , identify the linearized exterior differential system 2 2 with Im + P /P ⊂ P/P 2 = E ⊥ ⊗ Λ∗ E ∗ . In a tableau, P is generated by the polars, so the linearization is precisely the same tableau, modulo the nonlinearity. So the linearization about an involutive integral element is involutive. On the other hand, the linearization at a noninvolutive integral element may or may not be involutive. 6.5 Compute the linearization of uxx = uyy + uzz + u2x around u = 0 by setting up this equation as an exterior differential system. 6.6 For a section v of the normal bundle of X which vanishes at a point m, compute the linearization Lv ϑ|E .

The characteristic variety An integral element is noncharacteristic if it is a hyperplane in precisely one integral element. An integral manifold is noncharacteristic (or characteristic) if all of its tangent spaces are. A characteristic hypersurface in an integral manifold could perhaps lie in more than one integral manifold, a potential nontangential intersection of integral manifolds. Glue along such an intersection, to create a “crease” along an integral manifold. So we imagine that integral manifolds are more flexible along characteristic hypersurfaces, although there is no theorem to prove that. 6.7 Find the characteristics of the wave equation as an exterior differential system. Show creasing of solutions along characteristics, and not along noncharacteristic curves. Lemma 6.1. Take an integral manifold X of an exterior differential system I, a point x ∈ X, and let E ..= Tx X. The characteristic variety Ξx ⊂ PE ∗ of the linearization is the set of characteristic hyperplanes in E. Proof. We compute the symbol by replacing derivatives v a ξi : σ(ξ) v = ξ ∧ (v ϑ)|E .

∂v a ∂xi

by expressions

The linearization of I is just the sum of the linearizations of any forms ϑ ∈ I spanning Im : p∗ σ : ξ ∈ E ∗ , v ∈ Tm M/E 7→ Im ⊗ Λp E ∗ . So Ξx is the set of ξ ∈ E ∗ for which there is some section v of the normal bundle of X with v(x) 6= 0 and ξ ∧ (v ϑ)|E = 0 p for every ϑ ∈ Im . This says precisely that the vector v can be added to the hyperplane E ∩ (0 = ξ) ⊂ E to make an integral element enlarging the hyperplane.

Determined systems

This lemma allows us to define the characteristic variety of any integral element, even if not tangent to any integral manifold: the characteristic variety of an integral element E is the set of characteristic hyperplanes in E, denoted ΞE ⊂ PE ∗ . 6.8 For an involutive integral element, prove that the following are equivalent: a. the integral element contains a noncharacteristic hyperplane, b. every regular hyperplane is noncharacteristic, c. the final character is zero. 6.9 More equations, fewer characteristics: if J ⊂ I, prove that ΞI ⊂ ΞJ . 6.10 What is the characteristic variety of a Frobenius exterior differential system?

Determined systems A p-dimensional integral element E ⊂ Tm0 M of an exterior differential system I p is determined if E contains a noncharacteristic hyperplane and Im has constant dimension s0 + s1 + · · · + sp−1 for m near m0 . Note that then sp = 0. Theorem 6.2. Take an analytic exterior differential system. Every noncharacteristic analytic integral manifold X, whose every tangent space is a hyperplane in a determined integral element, is a hypersurface in an analytic integral manifold. Any two analytic integral manifolds containing X as a hypersurface share a set, open in both, containing X. Proof. As above, the symbol matrix σ(ξ) at each p-dimensional integral element E is a linear map in p∗ Tm M/E → Im ⊗ Λp E ∗ , p square at each determined integral element E. The dimension of Im cannot drop below s0 + s1 + · · · + sp , because it generates that many polar equations at E, and hence near E. Pick out that number of linearly independent p-forms from I: they span I p at every point nearby. The determined exterior differential system they generate has the same p-dimensional integral manifolds, polar equations, and symbol.

6.11 Prove that every real immersed curve in an almost complex manifold lies in an immersed holomorphic disk. 6.12 Prove that if E ⊂ E+ is characteristic, and E+ has dimension p then sp > 0. To compute the characteristic variety from a tableau: a. Drop the nonlinearity.

49

50

The characteristic variety

b. Turn each polar π α into a formal expression v α ξ where ξ = ξi ω i . c. Expand out the tableau, and collect up terms in each ω I . d. Write out these terms as linear expressions in the variables v α , and reorganize those linear expressions in a matrix: the symbol matrix σ(ξ) v = 0. e. The determinants of the minors of the symbol matrix are the equations of Ξ. This works because the equations of the characteristic variety include 0 = θa (v)ξ, which forces 0 = θa (v) for any nonzero ξ, so that the nonzero components of v are v α = π α (v). Compute ξ ∧ (v ϑ) by plugging in v to each polar π α , yielding v α , and wedge in a factor of ξ. For the tableau of triply orthogonal webs

we find  3 v 0 0

0 v2 0

π3

0

0

  

0

π2

0

0

0

π1

  ω 12    ∧  ω 13    ω 23

  12   3   0 ω v 0 0 ξ3 0  ξ ∧ ω 13  =  0 v 2 0  −ξ2  ω 123 , ξ1 v1 ω 23 0 0 v1   1  v 0 0 ξ3 =  0 −ξ2 0  v 2  ω 123 , ξ1 0 0 v3 = σ(ξ) vω 123 .

So det σ(ξ) = ξ1 ξ2 ξ3 : the characteristic variety Ξ is the triple of lines (ξ1 = 0), (ξ2 = 0) and (ξ3 = 0). Any 3-dimensional integral element coframed by ω 1 , ω 2 , ω 3 has characteristic hyperplanes 0 = ξ1 ω 1 + ξ2 ω 2 + ξ3 ω 3 for any one of the three coefficients vanishing, i.e. containing any one of the three axes. Theorem 6.3. Take an embedded analytic surface S ⊂ E3 . Pick orthonormal analytic vector fields e1 , e2 , e3 along S, none tangent to S. Then there is a triply orthogonal web near S in E3 with leaves perpendicular to e1 , e2 , e3 at each point of S. Any two such agree near S. The reader familiar with Riemannian geometry may recognize that this proof works identically replacing E3 by any analytic Riemannian 3-manifold. Any symmetry of S and e1 , e2 , e3 on S is shared by the triply orthogonal web, by uniqueness.

51

Determined systems

In proving Lie’s third theorem,  π1 π21  1  2 π22  π1   . ..  .. .  p π1 π2p

we had tableau  1 . . . πp   ω 1  2  ω 2  . . . πp   ∧  . , ..  .  .  ... .   ωp p . . . πp

i i i.e. 0 = πj ∧ ω j . We plug in πj = v α ξ, to get

0=

1 i (v ξk − vki ξj )ω kj . 2 j

Our linear expressions are vji ξk −vki ξj . One particular solution v to these equations is vji ..= ξj for all i. So every hyperplane in every p-dimensional integral element is characteristic. This is not surprising: any diffeomorphism takes every Maurer–Cartan form to a Maurer–Cartan form; we can pick a diffeomorphism which is very close to the identity except along a hypersurface “bending” every Maurer–Cartan form “along” that hypersurface. 6.13 Find the characteristic variety of ∇ × u = u − f . 6.14 Find the characteristic variety for the exterior differential system of harmonic functions in the plane. Show that it doesn’t satisfy the conditions of the Cauchy–Kovalevskaya theorem as described in theorem 6.2 on page 49. (Harder: can you “fix it”?)

Chapter 7

Proof of the Cartan–Kähler theorem

Cartan’s strategy II This section can be omitted without loss of continuity.

Pick a hypersurface, a hypersurface in the hypersurface, and so on, a flag of submanifolds, stopping at codimension p. Imagine that the generic p-dimensional integral manifold locally intersects each submanifold of our flag transversally: intersecting the smallest in a point, the next smallest in an integral curve, the next smallest in an integral surface, and so on. Construct an integral manifold by Cartan’s strategy: draw a point inside the smallest submanifold of the flag, an integral curve through that point inside the next smallest, an integral surface through that curve inside the next smallest, and so on. Trouble: there might be many integral curves, passing through that point, lying on that flag submanifold. Inside the flag submanifold, pick a smaller submanifold, a restraining manifold, cutting down dimensions so that our point lies on a unique integral curve in the restraining manifold. We choose nested restraining manifolds, one in each flag submanifold, starting from the largest and going down in dimension. Start from the smallest and go up in dimension: pick a point in the smallest restraining manifold, sweep out an integral curve through it in the next restraining manifold, and so on. We will see that each restraining manifold is locally the choice of sk functions of k variables. Fix a flag. Construct integral manifolds by varying the restraining manifolds inside the submanifolds of that fixed flag. We will see that different selections of restraining manifolds give rise to a different integral manifold in the final stage. In this sense, integral manifolds depend on s0 constants, s1 functions of one variable, and so on. We wrote out differential equations on page 42. The flag: Mi = (0 = xi+1 = · · · = xp ) the variables over which to solve differential equations. The restraining manifolds are the initial data: Rp = (up = up (x1 , . . . , xp )) and, for i = p − 1, p − 2, . . . , 2, 1, 0, Ri = Ri+1 ∩ Mi ∩ (ui = ui (x1 , . . . , xi , 0, . . . , 0)).

53

54

Proof of the Cartan–Kähler theorem

7.1 For the equation ∇ × u = f − u from problem 2.5 on page 21, write flag and restraining manifolds in coordinates, and the associated differential equations.

Chapter summary Theorem 7.1 (Cartan–Kähler I). Take an analytic exterior differential system and a noncharacteristic analytic integral manifold X with locally maximal rank polar equations in every tangent space. Then X is a hypersurface in an analytic integral manifold, locally unique in that any two analytic integral manifolds in which X is a hypersurface both contain a subset, open in both and containing X. A submanifold R of M restrains an integral manifold X of an exterior differential system if the exterior differential system pulls back to R to make X noncharacteristic, with each tangent space of X having locally maximal rank polar equations. Corollary 7.2. If an analytic manifold restrains an analytic integral manifold of an analytic exterior differential system, then the integral manifold is a hypersurface in a locally unique analytic integral submanifold of the restraining manifold. Take an integral element E ⊂ Tx M of an exterior differential system. A linear subspace V ⊂ Tx M restrains E if E ⊆ V and a. all nonzero polar equations of E, being linear functions on Tx M , pull back to nonzero linear functions on V and b. the vectors in V on which those polar equations vanish form a subspace containing E as a hyperplane. Theorem 7.3 (Cartan–Kähler II). Take an exterior differential system I on a manifold M and an integral manifold X with locally maximal rank polar equations. Take a submanifold R containing X so that, for each x ∈ X, Tx R restrains Tx X. Then R restrains X, corollary 7.2 applies. Proof. When we pull back to R, the polar equations of integral elements near Tx X also pull back without losing rank. So polar equations on Tx X have locally maximal dimension among all nearby integral elements tangent to R. At each x ∈ X, Tx X is a hyperplane in a unique integral element in Tx R: the vanishing locus of the polar equations. So X is noncharacteristic in R; apply theorem 7.1. Corollary 7.4. If the tangent spaces of an analytic integral manifold of an analytic exterior differential system have locally maximal rank polar equations, then the following are equivalent: a. that rank is less than the codimension of the integral manifold,

55

Extending an integral manifold

b. the integral manifold is covered in open sets, each of which is a hypersurface in an analytic integral manifold. Proof. In one tangent space Tx M , pick covectors, linearly independent modulo the polar equations of Tx X, so that the subspace on which they and the polar equations vanish is a hyperplane containing Tx X. That hyperplane can be any one that contains Tx X and satisfies the polar equations of Tx X, so in particular we can make Tx X noncharacteristic in it. Pick any submanifold R normal to those covectors.

Extending an integral manifold Proof of theorem 7.1 on the preceding page. Proof. Let p ..= 1 + dim X; denote X as Xp−1 . At each x ∈ Xp−1 , there is a unique p-dimensional integral element Ep ⊂ Tx M containing Ep−1 ..= Tx Xp−1 . By the solution of problem 6.8 on page 49, sp = 0 on Ep . By problems 5.7 and 5.8 on page 41, we can take a torsion-free tableau for I, defined in some open set, adapted to a generic flag which includes Ep−1 and Ep . Each polar ϑ = · · · + π ∧ ωI + . . . has I ⊆ { 1, 2, . . . , p − 1 }, since sp = 0. Let J ..= { 1, 2, . . . , p − 1 } − I, and ϑ0 ..= ϑ ∧ ω J . So ϑ0 is ϑ “raised up” to become a p-form of grade p − 1, shifting its polar to that grade. Similarly, if ϑ is one of the coframing 1-forms θa , let ϑ0 = θa ∧ ω 1...p−1 . Let I 0 be the exterior differential system generated by these p-forms ϑ0 . 7.2 Prove that Xp−1 is a hypersurface in some I 0 -integral manifold Xp . We need to see that I = 0 on Xp . The 1-forms ω 1 , . . . , ω p coframe Xp near x. Every p-form in I p on Xp is wedged up a form ϑ ∈ I by some ω J . If p ∈ / J, then ϑ ∧ ω J ∈ I 0 vanishes on Xp . So every p-form in I p on Xp is a multiple of ω p . We will see in lemma 7.6 on the following page that I = 0 on Xp .

Background material from differential geometry Lemma 7.5. For any (p + 1)-form ϑ and vector fields v0 , . . . , vp , dϑ(v0 , . . . , vp ) = (−1)i vi (ϑ(v0 , . . . , vˆi , . . . , vp )) X + (−1)i+j ϑ([vi , vj ], v0 , . . . , vˆi , . . . , vˆj , . . . , vp ). i y, and S is a Weingarten surface associated to W . On its frame bundle ⌜S in ⌜E3 , we have 

ω3 = 0,   γ13 a11 = γ23 a12

a12 a22

  ω1 , ω2

0 = a12 − a21 , K = a11 a22 − a212 , a11 + a22 , H= 2 (H, K) ∈ W. ˆ be the set of symmetric matrices Let W   a11 a12 a= a12 a22 so that



 tr a , det a ∈ W. 2 73

74

Example: Weingarten surfaces

ˆ of Then X ..= ⌜S is a 3-dimensional integral manifold in M ..= ⌜E3 × W     ω3 θ0 θ1  = γ13 − (a11 ω1 + a12 ω2 ) θ2 γ23 − (a21 ω1 + a22 ω2 ) on which ω1 , ω2 , ω12 are linearly independent. Calculate the tableau:    0 θ0  π 1 d θ1  = −   θ2 π2 where

     π1 a11 0 π2  = d a12  + −1 π3 a22 0

0

0

π2 π3

  ω1   ω2  0 ∧  γ12 0

2 0 −2

  a11 0 1 a12  ω12 . 0 a22

Locally, we can write W as the set of solutions of an equation f (x, y) = 0 in the plane with df 6= 0. So on S, 0=f



 tr a , det a . 2

Let fH ..=

∂f ∂f , fK ..= . ∂H ∂K

Compute out that this gives 0=



da11 + da22 2



fH + (a22 da11 + a11 da22 − 2a12 da12 ) fK .

In terms of π1 , π2 , π3 this relation is 0=



π1 + π3 2



fH + (a22 π1 + a11 π3 − 2a12 π2 ) fK

This equation has coefficients of π1 , π2 , π3 given by a11 fK + fH , a22 fK + fH , a12 fK . 10.2 Prove that all of these vanish, i.e. there is no addition linear relation among π1 , π2 , π3 , precisely when a11 = a22 and a12 = 0, i.e. an umbilic point. Since our curve W lies inside H 2 > K (“away from umbilic points”), our exterior differential system has characters s1 = 2, s2 = 0, s3 = 0 so involution.

75

Cauchy characteristics

Cauchy characteristics The Cauchy characteristics are the rotations of frame tangent to the surface. ¯ is the set of choices of point in E3 , The 6-dimensional quotient manifold M plane through that point, unit normal vector to that plane, and symmetric bilinear form on that plane, with half trace and determinant lying in W . On ¯ , the exterior differential system is determined. Each Weingarten surface S M gives an integral surface of that exterior differential system, mapping each point of S to its tangent plane and shape operator on that tangent plane.

Characteristic variety The symbol matrix is −ξ2 fH ξ + f (a 1 K 22 ξ1 − a12 ξ2 ) 2

! ξ1 fH 2 ξ2 + fK (a11 ξ2 − a12 ξ1 )

which has determinant  fH 2  ξ1 + ξ22 . −fK a11 ξ22 − 2a12 ξ1 ξ2 + a22 ξ12 − 2 Recall that the characteristic variety consists of the hyperplanes X 0= ξi ωi = 0, i

satisfying these equations. A vector v = v1 e1 + v2 e2 lies in such a characteristic hyperplane just when 0 = ξ1 v1 + ξ2 v2 , so then, up to scaling (ξ1 , ξ2 ) = (v2 , −v1 ) . Plug this in to see that the characteristics are   fH 2 v1 + v22 . 0 = fK a11 v12 + 2a12 v1 v2 + a22 v22 + 2 i.e. in classical notation,

fH I. 2 So the characteristics are the curves on S with velocity v satisfying this quadratic equation. Since we have assumed that our surface contains no umbilic points and that df 6= 0, not every curve is characteristic. 0 = fK II +

Initial data Take a ribbon along a curve C in E3 . At each point of that curve, draw the perpendicular plane to the ruling line of the ribbon. On that plane, take a

76

Example: Weingarten surfaces

symmetric bilinear form II with two distinct eigenvalues, analytically varying along C. Let H ..= tr II/2, K ..= det II. The form II is nondegenerate if, on every tangent line to C, the homogeneous cubic form dH II +

dK I 2

is not zero, where I is the Euclidean inner product. Consider what a noncharacteristic curve looks like in the 6-dimensional ˆ . Such a curve consists of a ribbon x(t), e3 (t), together manifold M = ⌜E3 × W with the additional data of e1 , e2 and the values aij . The curve x, e, a is an integral curve of the exterior differential system, i.e. ω3 = 0 and γi3 = aij ωj . The equation ω3 = 0 is just the requirement that e3 ⊥ x, ˙ i.e. a ribbon. Since we can rotate the frame e1 , e2 , we can ask that e1 be tangent to the curve x(t). So then γi3 = aij ωj just when de1 , dt de2 II(e1 , e2 ) = −e3 · . dt II(e1 , e1 ) = −e3 ·

So the shape operator is partly determined by the ribbon. Noncharacteristicity is precisely that fH de1 fK 6= . e3 · dt 2 10.3 Prove that the Weingarten equation f = 0 locally recovers the coefficient a22 , hence the entire shape operator, from the data of the ribbon. Parameterize a curve C by arc length as x(s), and let e1 ..= x. ˙ Take a ribbon on that curve and write the direction of the ruling line as e3 . Let e2 be the vector so that e1 , e2 , e3 is a positively oriented orthonormal frame along C. A form II is compatible with the ribbon if II(e1 , e1 ) = −e3 · e˙ 1 and II(e1 , e2 ) = −e3 · e˙ 2 . Compatibility is independent of the choice of arc length parameterization and of rotation of e1 , e2 . Define H and K, the trace and determinant of II. Define an immersed curve W : the image of the s 7→ (H(s), K(s)). Theorem 10.1. Take an analytic ribbon along a connected curve C in E3 , and a symmetric bilinear form II, defined in the perpendicular plane of the ruling line of the ribbon at one point of C, nondegenerate and compatible with the ribbon. Then II extends uniquely locally to be defined along an open subset of C, analytically varying, nondegenerate and compatible with the ribbon. If II extends to all of C, then there is an analytic Weingarten surface S in E3 containing C whose shape operator is II at each point of C and whose Gauss and mean curvature lie in the image of C in the plane under the map (H, K). Any two such surfaces agree near C. 10.4 Which analytic curves in E3 are geodesics on minimal surfaces?

Appendix A

The Cauchy–Kovalevskaya theorem

We prove the Cauchy–Kovalevskaya theorem: analytic determined systems of partial differential equations have local solutions, with arbitrary initial conditions.

Formal Taylor series For x1 , x2 , . . . , xn real variables and a1 , a2 , . . . , an nonnegative integers, let x ..= (x1 , x2 , . . . , xn ), a ..= (a1 , a2 , . . . , an ), xa ..= xa1 1 . . . xann , a! ..= a1 ! . . . an ! and ∂ a1 ∂ an ∂ a ..= . . . . ∂xa1 1 ∂xann P a ca (x − x0 ) with real constants ca , A formal Taylor series is an expression not required to converge. P A.1 Prove that the formal Taylor series n (2n)!tn diverges for t = 6 0. We add, subtract, multiply, differentiate and compose in the obvious way: finitely many terms at a time. Crucially, each output term depends only on input terms of lower or equal order (or, for the derivative, on just one order higher), so on only finitely many input terms. When we add, multiply, differentiate or compose, each step is only adding or multiplying coefficients. In particular, the sum, product, derivative (in any variable) and composition of formal Taylor series with positive terms P has positive terms. P A formal Taylor series ba xa majorizes another ca xa if ba ≥ |ca | for all a. If a convergent series majorizes another, the other is absolutely convergent. If f majorizes g and u majorizes v then f ◦u, f +u, f u, ∂ af majorizes g◦v, g+v, gv, ∂ ag respectively. The geometric series 1 = 1 + t + t2 + . . . 1−t has a Taylor series with positive coefficients. A formal Taylor series c0 +c1 t+. . . with all |cj | < 1 converges absolutely for |t| < 1, because the geometric series 77

78

The Cauchy–Kovalevskaya theorem

converges and majorizes it. Rescaling t and our series, we see that if a formal Taylor series is majorized by a geometric series, then it converges near the origin, i.e. if |cj | are bounded by a geometric series, in other words |cj | ≤ M rj for some M, r > 0, then c0 + c1 t + . . . converges absolutely near the origin. An analytic function is one which is locally the sum of a convergent Taylor series. Lemma A.1. A formal Taylor series converges to an analytic function just when it is majorized by a product of geometric series in one variable each. Proof. We give the proof for one variable around the point x = 0, and let the reader generalize. Take any analytic function f (x) with convergent Taylor series P P f (x) = cn xn . Since it converges absolutely for x near 0, |cn | rn converges for r near 0, so the terms of this series are bounded. Rescale to get a bound of 1, i.e. |cn | rn < 1 for all n. Therefore |cn | ≤

1 rn

i.e. f (x) is majorized by 1/(1 − x). A.2 Prove that any two formal Taylor series which converge near the origin to the same analytic function agree. A.3 Prove that an analytic function given by a convergent Taylor series around some point is also given by a convergent Taylor series around every nearby point. A.4 Prove that any two analytic functions defined on a connected open set which agree near some point agree everywhere. The derivative and composition of analytic functions is analytic, and the analytic inverse and implicit function theorems hold with the usual proofs [24, 31].

Complexification Every Taylor series converging in some domain of real variables continues to converge for complex values of those variables, with small imaginary parts, by the same majorization. Hence every analytic function of real variables extends to an open set in a domain of complex variables. By elementary complex analysis [1], sums, differences, products and derivatives of analytic functions on an open set are analytic, as are quotients, wherever the denominator doesn’t vanish, and the analytic inverse and implicit function theorems hold.

Solving for Taylor series using a differential equation Let’s solve a simple partial differential equation ut = ux . Think about the vector field X = ∂t − ∂x in the tx-plane.

Solving for Taylor series using a differential equation

Clearly Xu = 0 just when u satisfies our equation; but this happens just when u is constant along the flow lines of X. The flow lines of X are the diagonal lines, i.e. t + x constant, so the solutions of our partial differential equation are u(t, x) = U (t + x) for any function U . Solutions of partial differential equations can involve arbitrary functions, as in this example. Each solution u(t, x) is determined by its value at “time” t = 0. If we pick our initial data function U not smooth at a point, then u(t, x) is not smooth along the flow line of that point. So solutions can be worse than the initial data from which we construct them. A.5 Suppose that X is an analytic vector field on a manifold M and that H ⊂ M is an embedded analytic hypersurface and X is not tangent to H at any point of H. Pick analytic functions v : H → R and f : M × R → R. Prove that there is an analytic function u : open ⊂ M → R so that u(m, 0) = v(m) for m ∈ H and Xu(m) = f (m, u(m)) wherever u is defined. Prove that any two such functions u = u1 and u = u2 agree near H. Take the differential equation ut = u2 + t. Differentiate both sides: utt =2u ut + 1,  =2u u2 + t + 1. Our differential equation allows us to replace differentiation in t with some expression without any derivatives. The differential equation ut = u2 uxx + sin(u) allows us, using the chain rule, to replace any number of t derivatives by complicated expressions in x derivatives. We can calculate any Taylor coefficient in any power of t as a function of finitely many Taylor coefficients involving only powers of x, i.e. Taylor coefficients of u(x, 0). Danger: after we compute all of the Taylor coefficients, we have no guarantee that they sum up to a convergent Taylor series. A formal solution of an analytic differential equation is a formal Taylor series so that expanding out the compositions term by term, all terms in the equation turn to zeroes. As above, for any formal Tayor series u(0, x), there is an associated formal Taylor series solution u(t, x) of each partial differential equation ut = f (t, x, u, ux , uxx , . . . , ux...x ) for any smooth function or formal Taylor series f .

79

80

The Cauchy–Kovalevskaya theorem

The heat equation ut = uxx has a unique formal Taylor series solution with intial condition 1 , u|t=0 = 1 + x2 given by ∞ X tj x2k u(t, x) = (−1)j+k (2(j + k))! j! (2k)! j,k=0

which doesn’t converge anywhere except at t = 0.

The method of majorants Theorem A.2 (Cauchy–Kovalevskaya). An analytic system of partial differential equations of the form ut = f (t, x, u, ux ), defined in an open set of t ∈ R, x ∈ Rn , u ∈ Rp , with any analytic initial conditions u = U (x) at t = t0 , has an analytic solution near t = t0 . Any two solutions agree near t = t0 . Proof. Instead of looking at Taylor series of solutions, look at Taylor series of equations. Start with a simpler problem: take a differential equation ut = f (t, u) with initial condition u(0) = u0 . Go back to our expressions for Taylor coefficients: ut (0) = f (0, u0 ) , utt (0) = ft (0, u0 ) + fu (0, u0 ) ut , = ft (0, u0 ) + fu (0, u0 ) f (0, u0 ) , uttt (0) = . . . . If u0 ≥ 0 and all of the Taylor coefficients of f are ≥ 0 then inductively all of the Taylor coefficients of u are ≥ 0. In other words, if u0 ≥ 0 and f majorizes 0 then u majorizes 0. By the same reasoning, if we have two differential equations ut = f (t, u) and vt = g(t, v) with initial conditions u(0) ≥ v(0), and if f majorizes g, then by the same induction, u majorizes v. In particular, if the Taylor series of u converges, then so does that of v. A.6 Check that the function u(t) = 1 −

p 1 + 2 log(1 − t)

satisfies the toy equation ut =

1 (1 − t)(1 − u)

with initial condition u = 0 at t = 0.

81

The method of majorants

Don’t expand the unpleasant function u(t) but expand the toy equation ut =

1 , (1 − t)(1 − u)

= 1 + t + t2 + . . . X = tj uk :



 1 + u + u2 + . . . ,

jk

a geometric series. Suitable rescaling of t and u produces any convergent geometric series we like. By lemma A.1 on page 78, any analytic function f (t, u) is majorized by a geometric series in t, u. So the equation ut = f (t, u) is majorized by the toy equation, after some rescaling of variables. So the solution u(t) to ut = f (t, u) with u(0) = 0 is majorized by the toy solution, so has convergent Taylor series. We will generalize this toy example several times, to get a larger class of examples of majorizing equations. The function p u(t, x) = 1 − x − (1 − x)2 − 2t satisfies ut =

1 + ux 1−x

with u = 0 at t = 0. With suitable rescalings of x, t, u, this equation majorizes any equation of the form ut = f (x, u) + g(x, u) ux for any analytic f, g, with x, u ∈ R in a suitable open set. Therefore all such equations have local analytic solutions so that u = 0 when t = 0. To allow more x variables: if u(t, s) is the function above p u(t, s) = 1 − s − (1 − s)2 − 2t P and we let v(t, x) = u(t, s) where we let s = i xi where x ∈ Rn , then P 1 + n1 vx i P vt = 1 − xi and v = 0 at t = 0. Again, with a little rescaling, this equation majorizes any equation of the form ut = f (x, u) + g(x, u) ux for any analytic f, g, and t, u ∈ R, x ∈ Rn in some open set. Of course the same trick works even if we allow many u functions, i.e. u ∈ Rp . To allow the t variable to enter into the right hand side, we add a new equation vt = 1 with initial condition v = 0 at t = 0, so we are just forcing v = t everywhere. Then any system like ut = f (t, x, u) + g(t, x, u)ux with u = 0 when t = 0 is equivalent to ut =f (v, x, u) + g(v, x, u)ux , vt =1,

82

The Cauchy–Kovalevskaya theorem

with u = v = 0 when t = 0. We have already seen that such systems have solutions near the origin, If we want to solve a system of the form ut = f (t, x, u, ux ), invent a new variable v and solve instead ut = f (t, x, u, v) , vt = fx (t, x, u, v) + fu (t, x, u, v) v + fv (t, x, u, v) vx , with u = v = 0 when t = 0, to force v = ux . Suppose that we want to solve a system ut = f (t, x, u, ux ) with u = U (x) at t = 0 instead of u = 0 at t = 0. Replace the system by the system vt = f (t, x, v+U, vx +Ux ) with v = 0 at t = 0 and then let u(t, x) = v(t, x)+U (x). Given a solution u to the wave equation utt = uxx with initial conditions u = U and ut = W at t = 0, let Z Z t v(t, x) = W (x) dx + ux (s, x) ds 0

so that vt = ux . Differentiate under the integral sign and apply the fundamental theorem of calculus to find that vx = ut . Therefore u, v satisfy ut = vx , vt = ux with initial conditions u = U and v = W at t = 0. Conversely, any solution to this system of equations recovers our solution u to the wave equation. An analytic solution of the wave equation is uniquely determined by the initial height and velocity of the waves. To solve a second order system utt = f (t, x, u, ux , ut , uxx , uxt ) , with initial conditions u = U (x) and ux = W (x) at t = 0, let p = ux and q = ut . Then ut = q, pt = qx , qt = f (t, x, u, p, q, px , qx ) , with initial conditions u = U (x), p = U 0 (x), q = W (x) at t = 0. Any solution or formal solution of this system arises from a solution or formal solution to the original system and vice versa. This same trick reduces any system of any order to a first order system in more variables.

83

The method of majorants

Is there a vector field u so that ∇ × u = f for a given vector field f in E3 ? Write u = u1 , u2 , u3 and f = f 1 , f 2 , f 3 and let u1x2 ..=

∂u1 , etc. ∂x2

Our equation ∇ × u = f expands out to u3x2 − u2x3 = f 1 , u1x3 − u3x1 = f 2 , u2x1 − u1x2 = f 3 : 3 first order equations for 3 unknowns. By analogy with the Cauchy– Kovalevskaya theorem, we expect to find a unique solution with initial values of u1 , u2 , u3 given on some surface in E3x1 ,x2 ,x3 . This expectation is not correct. Taking ∇· on both sides of our equations reveals 0 = ∇ · f . If 0 6= ∇·f then there is no solution u. If 0 = ∇·f then the solutions are u = u0 + ∇φ for any one fixed solution u0 and for any twice continuously differentiable function φ. So the local solutions depend on one function φ of 3 variables, not on initial data from a surface in E3 . We might spot trouble on the horizon when we notice that we can’t solve our equations for the derivatives with respect to x1 , since there are only two of them, and similarly for x2 and for x3 . Nonetheless, we can easily make more complicated examples of systems of partial differential equations to which the Cauchy–Kovalevskaya theorem does not apply, with complicated “compatibility conditions” needed to ensure existence of a solution. Is there a vector field u so that ∇ × u = f − u for a given vector field f in E3 ? Our equation ∇ × u = f − u expands out to u3x2 − u2x3 = f 1 − u1 , u1x3 − u3x1 = f 2 − u2 , u2x1 − u1x2 = f 3 − u3 : 3 first order equations for 3 unknowns. Again we might expect to find a unique solution with initial values of u1 , u2 , u3 given on some surface in E3x1 ,x2 ,x3 . Again this intuition is not correct. Taking ∇· on both sides of our equations reveals 0 = ∇ · f − ∇ · u, i.e. u1x1 + u2x2 + u3x3 = fx11 + fx22 + fx33 , a fourth first-order equation which is linearly independent of the other three.

84

The Cauchy–Kovalevskaya theorem

Split the four equations as: u3x2 − u2x3 = f 1 − u1 , and u1x3 − u3x1 = f 2 − u2 , u2x1 − u1x2 = f 3 − u3 , u1x1 + u2x2 + u3x3 = fx11 + fx22 + fx33 . Solve the first of these equations: u3x2 − u2x3 = f 1 − u1 , but just along the surface x1 = 0, using the Cauchy–Kovalevskaya theorem in the variables x2 , x3 , starting with any choice of functions u1 , u2 on the surface x1 = 0 and any function u3 on the curve x1 = x2 = 0. Apply the Cauchy–Kovalevskaya theorem to the last 3 equations: u1x3 − u3x1 = f 2 − u2 , u2x1 − u1x2 = f 3 − u3 , u1x1 + u2x2 + u3x3 = fx11 + fx22 + fx33 . in all 3 variables x1 , x2 , x3 , by treating x1 as the variable we differentiate in:    1 −u2x2 − u3x3 + ∇ · f u  u2  =   u1x2 + f 3 − u3 ,  . u3 x u1x3 − f 2 + u2 1 The tricky question is whether the resulting functions actually solve the original problem. In fact, they do: we have solved the first equation u3x2 − u2x3 = f 1 − u1 , at x1 = 0. But if we differentiate that expression in x1 , check that  u3x2 − u2x3 − f 1 + u1 x = 0 1

modulo the 3 other equations, so vanishes by construction. The general solution to ∇ × u = f − u is given by picking one arbitrary function of one variable, and 2 arbitrary functions of two variables, again not what we would guess from familiarity with the Cauchy–Kovalevskaya theorem. Again, we might spot trouble on the horizon when we notice that we can’t solve our equations for the derivatives with respect to x1 , since there are only two of them, and similarly for x2 and for x3 .

Smooth counterexamples

Smooth counterexamples For functions which are not analytic, there are numerous theorems to prove the existence of solutions, under a wide variety of hypotheses, but there are some surprising counterexamples: a. The Cauchy–Riemann equations admit only analytic solutions: no solution has smooth nonanalytic initial data. b. Some linear determined equations Lu = f with analytic L have no solutions for any smooth f unless f is analytic [25].

85

Appendix B

Symbol and characteristic variety

We define the notion of symbol of a system of partial differential equations.

The problem The Cauchy–Kovalevskaya theorem says that we can solve a system of differential equations, if we can write it as solving for the highest derivative in a single variable t in terms of all other derivatives, in all variables, and the values of independent and dependent variables. If we can’t do this in the variables we are given, we might be able to do it after a change of variables. So we need a coordinate free description of “how many derivatives” are taken in a given direction. The Laplace operator in the plane: u 7→ uxx + uyy . At first glance, it appears to take two derivatives along each coordinate axis, and no derivatives in any other directions. But the Laplace operator is invariant under rotation, so it actually “feels” the second derivatives in all directions.

The symbol of a linear operator If you differentiate a high frequency wave in a direction in which it oscillates, it gets bigger. If you differentiate in a direction in which the wave is constant, it gets killed. We detect derivatives by plugging in waves. Take a large number λ and any function f (x) vanishing at the origin. Near iλf the origin, the function looks like a high frequency wave: expand f in a P e Taylor series f (x) = ξi xi + . . ., eiλf = eiλ

P

87

ξi xi +...

.

88

Symbol and characteristic variety

Any linear differential operator, P =

X

ca (x)∂ a

|a|≤m

is a polynomial P = P (x, ∂) in the operators ∂xi ; the coefficients are functions of x. Write this polynomial X P (x, ξ) = ca (x)ξ a . |a|≤m

Let Pm be the highest order terms, also called the symbol: X Pm (x, ξ) = ca (x)ξ a , |a|=m

also denoted σP (ξ), for ξ = ξi dxi ∈ T ∗ M a 1-form. To see “how many derivatives” P takes in some direction, expand: X    e−iλf P eiλf u = e−iλf ca (x)∂ a eiλf u , |a|≤m

Every time a derivative hits eiλf u, it either hits the exponential factor, pulling down a power of λ, or hits the u, so no power of λ. So the leading order term in λ is X   e−iλf P eiλf u = im λm ca (x)ξ a u + O(λ)m−1 , |a|=m

= i λ Pm (x, ξ)u + O(λ)m−1 . m m

where ξ = df (x). Without taking coordinates,   e−iλf P eiλf u σP (df ) u = lim λ→∞ im λm for any continuously differentiable function f vanishing at x. The heat operator in the plane is P u = ut − uxx − uyy

Background material from projective geometry

for a function u = u(t, x, y), and the symbol is σP (c dt + a dx + b dy) = −a2 − b2 . All along we could have allowed u to be valued in, say, Rq , and allow the coefficients ca (x) to be p × q matrices. Then σP (ξ) is a p × q matrix. The matrix entries are functions of x and polynomials of degree m in ξ. B.1 Find the symbol of the Cauchy–Riemann operator     ux − vy u P = . v uy + vx B.2 If we are interested in a linear system of equations 0 = P u, rather than an operator P , the equations have the same solutions if we rescale both sides by some invertible matrix g(x). Similarly, we could rescale the u variable by a matrix h(x). Check that σgP h (ξ) = gσP (ξ) h. P ∂y If we change variables by a diffeomorphism y = h(x), then ∂xi = j ∂xji ∂yj P P ∂y and for any 1-form ξ = j ξj dyj = ij ξj ∂xji dxi , so when we substitute ξi for ∂yi , σh∗ P (ξ) = σP (h∗ ξ) . We knew this already: the symbol σP (ξ) is defined independently of coordinates. Taking exterior derivative d as our differential operator, e−iλf d(eiλf ) = iλdf ∧ ω + dω, so the symbol is

σd (ξ) = ξ∧,

i.e. the symbol is the linear transformation ω 7→ ξ ∧ ω, where ξ is a 1-form.

Background material from projective geometry The projective space PV of a vector space V , over any field, is the set of lines through the origin of V [29]. If V has dimension n + 1, we may write PV as Pn . For any v ∈ V with v 6= 0, [v] ∈ PV is the line through v. Any nonzero linear function f on V vanishes on a hyperplane in V . This hyperplane determines f up to rescaling, and vice versa. So each point [f ] of PV ∗ is naturally identified with a hyperplane in V .

89

90

Symbol and characteristic variety

The characteristic variety The characteristic variety of a linear differential operator P is ∗ ΞP,m = { [ξ] ∈ PTm M | ker σP (ξ) 6= 0 } . C∗ ∗ Let Tm M ..= Tm M ⊗ C. The complex characteristic variety consists of complex lines in a complex vector space:  C C∗ ΞP,m = [ξ] ∈ PTm M ker σP (ξ) 6= 0 .

A 1-form ξ belongs to the characteristic variety just when the operator P takes “exceptionally few” derivatives in the direction of ξ. For the heat operator, the characteristic variety is the set of [c, a, b] ∈ P2 so that a2 + b2 = 0, i.e. the single point [c, a, b] = [1, 0, 0] ∈ P2 , while complex characteristic variety consists of the complex number solutions to the same equations: [c, a, ±a], i.e. b = ia and b = −ia, a union of two complex lines. Take exterior derivative d on k-forms as our differential operator. The symbol is σd (ξ) = ξ∧, so the characteristic variety is the set of all lines [ξ] spanned by 1-forms ξ 6= 0 so that ξ ∧ ω = 0 for some ω 6= 0. If we work with only 0-forms ω, then this forces ξ = 0: the characteristic variety is empty. If k > 0, then any ξ has ξ ∧ ω = 0 for some ω 6= 0: take any ω of the form ξ ∧ η. So the characteristic variety of the exterior derivative on k-forms is ( ∅, if k = 0, Ξd,m = ∗ PTm M, if k > 0. The same calculation computes the complex characteristic variety: ( ∅, if k = 0, C Ξd,m = C∗ PTm M, if k > 0. For the heat operator, the characteristic variety at each point consists of the single hyperplane dt = 0. This hyperplane is the tangent plane to each surface representing space at constant time. For a vector field X, thought of as a differential operator P u = Xu, the characteristic variety is the set of [ξ] so that ξ(X) = 0, i.e. the set of all hyperplanes containing the vector X.

91

Linearization

For the wave operator, P u = utt − uxx − uyy , the symbol is σP (t, x, y, c, a, b) = c2 − a2 − b2 . Up to rescaling, we can arrange c = 1, so the characteristic variety is a circle [1, a, b] ∈ P2 so that a2 + b2 = 1. As a family of hyperplanes, this circle is the set of hyperplanes tangent to the “light cone” dt2 = dx2 + dy 2 . Points of the complex characteristic variety are complex hyperplanes in the spaces of complexified tangent vectors. B.3 Find the characteristic variety of Maxwell’s equations in the vacuum.

Linearization Take a nonlinear differential equation, say ut = uxx u + sin(ux ), and take a solution u(t, x). Take a function v(t, x) and plug in u + εv instead of u to both sides, supposing that u + εv is also a solution, at least up to an error smaller than ε: ut + εvt = (uxx + εvxx ) (u + εv) + sin(ux + εvx ) , =uuxx + sin(ux ) + ε (uxx v + vxx u + cos(ux ) vx ) + O(ε)2 using the Taylor series of sin. Since u satisfies the differential equation, we can cancel off ut = uuxx + sin(ux ) from both sides, and divide by ε and send ε → 0: vt = uxx v + vxx u + cos(ux ) vx , a linear differential equation in v, but with coefficients which depend on the solution u that we perturbed about. More generally, for any expression P [u] = F (x, u, ux ) (a differential operator, perhaps nonlinear), and any function u for which P [u] = 0, the linearization of P about u is the linear differential operator P 0 [u]v = Fu (x, u, ux ) v + Fux (x, u, ux ) vx , and similarly for higher order operators. B.4 Let ∆ = ∂xx + ∂yy . Linearize the equation ∆u = |du|2 around the solution u = 1. B.5 Let ∆ = ∂xx + ∂yy. Linearize the Liouville equation ∆ log u = −u2 around the solution u = 12 log (1−x24−y2 )2 . When we take about a differential equation 0 = F (x, u, ux ), defined for x ∈ Rn , u ∈ Rk , ux ∈ Rnk lying in some open set (x, u, ux ) ∈ U ⊂ Rn+k+nk , we

92

Symbol and characteristic variety

implicitly assume that F is analytic and that the set M of all (x, u, p) ∈ U ⊂ Rn+k+nk where F (x, u, p) = 0 is a manifold and that the map (x, u, p) ∈ M 7→ (x, u) is a submersion (so the differential equation doesn’t, locally, constraint the value of the independent or dependent variables). If we are given the values of u and ux at just one point x, then we can calculate F (x, u, ux ) at that one point, and if 0 = F (x, u, ux ) then we can also calculate the linearization P 0 [u] at x, or in other words we can calculate the coefficients of the linearization at x: Fu (x, u, ux ) and Fux (x, u, ux ) . Therefore we can calculate the characteristic variety Ξx,u,ux of our differential equation, by which we mean that of the linearization at (x, u, ux ). The linearization of ut = uuxx + uyy + u2 is vt = vuxx + uvxx + vyy + 2uv, so the characteristic variety is 0 = 0 + uξx2 + ξy2 + 0, if we write the components of a 1-form ξ as ξ = ξt dt + ξx dx + ξy dy.

Initial value problems A differential equation is determined just when the symbol matrix σ(ξ) is square, and invertible for some ξ. Lemma B.1. A differential equation 0 = F (x, u, ux ) can locally be written in the form of the Cauchy–Kovalevskaya theorem in some coordinates just when it is determined. Proof. If we linearize the equation around some point (x, u, p), we get the linear equation 0 = Fu (x, u, ux )v + Fp (x, u, ux )vx , with symbol σ(ξ) v = Fp (x, u, ux )ξv. So if we write coordinates as xi , ua , pai , then the symbol matrix is σ(ξ) = Fpi ξi . We want to see when we can somehow smoothly solve, at least locally, the equation 0 = F (x, u, ux ) for some uxi as a function of x, u and the other uxj . By the implicit function theorem, we can do this just when Fpi is an invertible square matrix. Note that σ(dxi ) = Fpi . On the other hand, if σ(ξ) is an invertible square matrix for some ξ, we can linearly change variables to arrange that ξ = dxi and reverse our steps. The problem of specifying initial data for nonlinear equations is much more complicated than for linear equations. Take a hypersurface H ⊂ Rn of the x-variables and maps u(x) and p(x) defined along H so that p(x)v = u0 (x)v for

93

Initial value problems

any tangent vector v to H. The maps u(x), p(x) are noncharacteristic initial data if (x, u(x), p(x)) ∈ M for each x ∈ H and Tx H is noncharacteristic for the the linearization of our equation at each point (x, u(x), p(x)). Lemma B.2. A determined system of equations has some analytic noncharacteristic initial data through any point (x, u, p). Proof. Since the symbol matrix is somewhere invertible, the characteristic va∗ riety Ξm is not all of PTm M , where m = (x, u, p). Theorem B.3. Take a determined system of differential equations and noncharacteristic initial data along an embedded hypersurface, all analytic. There is an analytic solution to the differential equations agreeing with the initial data along the hypersurface. Any two such agree near the hypersurface and agree in any connected open set where both are defined. Proof. If we can prove uniqueness of local solutions, then we can glue them together to get a unique global solution. So it suffices to prove the result locally. By an analytic change of variables, arrange that our hypersurface is t = 0 in some (t, x) variables. By the implicit function theorem, if we have a smooth map F (t, x, u, p, q), we can locally smoothly solve F (t, x, u, p, q) = 0 for a variable p as a function of the other variables just when Fp is a matrix of full rank. We can rewrite the equation F (t, x, u, ut , ux ) = 0 in the Cauchy–Kovaleskaya form, i.e. solve for ut as a function of the other variables, just when Fut is a matrix of full rank, i.e. when t = 0 is not characteristic for the linearization of the differential equation. Corollary B.4. A determined analytic system of differential equations has a local analytic solution near any point. Take the equation u2t + u2x = 1 with initial data u = 0 at t = 0. Differentiating the requirement that u = 0, we require ux = 0 too at t = 0, as part of our initial data. The differential equation then says that ut = ±1 at t = 0. The linearization at any point is 2ut vt + 2ux vx = 0. Along our initial data, this is 2vt = 0. The linearization is determined, so the nonlinear equation is, and our theorem guarantees a solution near t = 0. Easier: we could have just solved p ut = ± 1 − u2x without using the theorem above, and then any initial data u(0, x) with −1 < ux < 1 all along t = 0 would give rise to a unique solution u(t, x) near t = 0.

94

Symbol and characteristic variety

Higher order differential equations Consider a linear differential operator P u = uxx ; its symbol is σP (ξ) = −ξ 2 . Take a first order operator with the same kernel:         u vx 0 1 u Q = = ∂ . v ux 1 0 x v The symbol of Q is



0 ξ

 ξ . 0  The characteristic variety is ΞQ = ξ 2 = 0 = ΞP . σQ (ξ) =

If there are many functions u of many variables x, this σP is a matrix-valued quadratic form, and the same steps tell us that ΞQ = ΞP and ΞQ (C) = ΞP (C): replacing a system of differential equations by its equivalent first order system doesn’t change the characteristic variety. The minimal surface equation   1 + u2x uyy − 2ux uy uxy + 1 + u2y uxx = 0, representing surfaces which have locally least area with given boundary, has empty characteristic variety at every point. If we pick any embedded real curve in the plane, and along that curve pick functions f, p, q, there is a unique solution near that curve so that u = f, ux = p, uy = q along the curve. B.6 Find the characteristic variety of the Euler–Tricomi equation utt = tuxx . B.7 If P = ∂xx + ∂yy , then the associated equation P u = 0 is the equation of an electrostatic potential energy u: the Laplace equation, and ∆ ..= P (D) is the Laplace operator. Show that the characteristic variety is cut out by the equation 0 = ξ 2 + η 2 , which has (ξ, η) = (0, 0) as its only solution. B.8 Calculate the linearization, symbol and complex characteristic variety of the minimal surface equation around a solution u(x, y). B.9 Suppose that 0 = F (x, u, ux ) is an analytic system of partial differential equations, and the symbol of the linearization about any point (x, u, ux ) has a covector ξ for which σ(ξ) is a surjective linear map, with kernel of rank k say. Prove that we can locally add k additional equations to make the system determined, and conclude that there are local solutions. What is the correct notion of noncharacteristic initial data?

Counterexamples

Counterexamples Linear analytic scalar equations which are not determined fail to have either existence or uniqueness of solutions, with analytic initial data [23]. For functions which are not analytic, uniqueness can fail even for determined equations: a. Some smooth linear determined equations have smooth solutions which agree on one side of a noncharacteristic hypersurface and disagree on the other [2]. b. Some analytic nonlinear determined equations have smooth solutions which agree on one side of a noncharacteristic hypersurface and disagree on the other [28]. There are some results ensuring analytic local solutions of analytic differential equations with initial data characteristic at some points [11].

95

Appendix C

Analytic convergence

Convergence A sequence f1 , f2 , . . . of analytic functions of real variables, defined on some open set, converges to an analytic function f just when that set lies in an open set in complex variables to which f and all but finitely many fi extend and where fi converge uniformly to f on compact sets. By the Cauchy integral theorem [1] p. 120, convergence is uniform with all derivatives on compact sets. C.1 Prove: a sequence f1 , f2 , . . . of analytic functions converges to an analytic function f just when we can cover their domain in compact sets, and cover each compact set by an open set in complex variables, to which f and all but finitely many fi extend and are bounded, and on which fi converge uniformly to f .

Commutative algebra A graded ring is a ring R which is the direct sum of abelian groups R0 , R1 , . . . so that Ri Rj ⊆ Ri+j , elementary if generated by R0 and R1 . An R-module M is graded if a direct sum of abelian groups Mi with Ri Mj ⊆ Mi+j ; asymptotically simple if R1 Mk = Mk+1 for all sufficiently large k; elementary if each Mk is finitely generated as an R0 -module. Lemma C.1 (Phat Nguyen [30]). An elementary graded module over an elementary graded ring is finitely generated if and only if asymptotically simple. Proof. Suppose M is finitely generated over R, hence by M0 + M1 + · · · + Mk . For any j ≥ k, Mj+1 = Rj+1 M0 + Rj M1 + · · · + R1 Mj . Since R is elementary Mj+1 = R1j+1 M0 + R1j M1 + · · · + R1 Mj ⊆ R1 Mj . But R1 Mj ⊆ Mj+1 , so Mj+1 = R1 Mj for all j ≥ k. Therefore M is asymptotically simple. Conversely, assume M asymptotically simple: R1 Mj = Mj+1 for any j ≥ k. As an R-module, M is generated by M0 + M1 + · · · + Mk . Because M is elementary, we can select a finite set of generators from each of M0 , . . . , Mk as an R0 -module, hence as an R-module. 97

98

Analytic convergence

A module M over a ring R is Noetherian if every R-submodule is finitely generated; R is Noetherian if every ideal is finitely generated. Theorem C.2 (Krull Intersection Theorem [30]). Suppose that R is a Noetherian commutative ring with identity, I ⊆ R an ideal, M a finitely generated R-module, and ∞ \ I j M. N ..= j=0

Then there is an element i ∈ I so that (1 + i)N = 0. If all elements of 1 + I are units of R, then N = 0. Proof. Write R, I, M, N as R0 , I0 , M0 , N0 . Define elementary graded ring and modules: R ..= R0 ⊕ I0 ⊕ I02 ⊕..., . M .= M0 ⊕ I0 M0 ⊕ I02 M0 ⊕ . . . , N ..= N0 ⊕ N0 ⊕ N0 ⊕ . . . . Since M is asymptotically simple, M is finitely generated over R. Since R0 is Noetherian, R is Noetherian, so M is Noetherian as an R-module. Because N0 ⊂ I0j M0 for j = 1, 2, . . ., N is a graded R-submodule of M , so finitely generated over R. By lemma C.1 on the previous page, N is asymptotically simple, so I0 N0 = N0 . P Take generators na ∈ N0 . Then na = b jab nb for some jab ∈ I0 . Let J = (jab ), a matrix, and n = (na ) a vector. So (I − J)n = 0. Multiply by the adjugate matrix (the transpose of the matrix of cofactors) of I − J to get det(I − J)n = 0; set 1 + i = det(I − J).

Germs The germ of an analytic function at a point is its equivalence class, two functions being equivalent if they agree near that point. A sequence of germs converges just when it arises from a convergent sequence of analytic functions. By majorizing, this is just when the Taylor coefficients converge to those of an analytic function. Germs form a ring, under addition and multiplication of functions. Pick a finite dimensional real vector space. Germs of analytic maps to that vector space form a module over that ring. Theorem C.3 (Hilbert basis theorem [8] p. 81 theorem 2.7, [18] p. 161 lemma 6.3.2, theorem 6.3.3, [34] p. 44 theorem 3.4.1). Every submodule of that module is finitely generated. Theorem C.4 (Henri Cartan [5] p. 194 corollaire premiere, [12] p. 46, [15] p. 85 theorem 3, [34] p. 274 theorem 11.2.6). Every submodule of that module is closed under convergence.

Germs

Proof. Pick a submodule S. Take a convergent sequence fi → f of map germs, with fi ∈ S. We need to prove that f ∈ S. Let R be the the ring of analytic function germs at the origin, I ⊂ R the germs vanishing at the origin, T the Rmodule of analytic map germs, which we identify with their Taylor series. The k-th order Taylor series are the elements of T /I k+1 T , and form a vector space of finite dimension, containing a linear subspace S/I k+1 S, which is therefore a closed subset, so contains the k-th order Taylor coefficient of f . Let M ..= T /S, so f ∈ I k M for all k = 1, 2, . . .. The set 1 + I is the set of germs of functions equal to 1 at the origin, so units in R. By Krull’s intersection theorem, f = 0 in M , i.e. f ∈ S.

99

Appendix D

The moving frame

In most chapters of these notes, we study geometry that is invariant under diffeomorphisms. In this chapter, we study geometry that is invariant under rigid motions of 3-dimensional Euclidean space. We want examples of exterior differential systems. For simplicity, we look to surface theory: differential equations governing surfaces in 3-dimensional Euclidean space. Naturally, we demand that our differential equations be invariant under rigid motion. But there are no differential forms on Euclidean space invariant under rigid motion. We will instead find invariant differential forms on the frame bundle of Euclidean space.

Notation Warning: We use the notation E3 instead of R3 to mean Euclidean space, i.e. R3 with its usual metric. Warning: In this chapter, e1 , e2 , e3 denotes an arbitrary orthonormal basis of E3 , not necessarily the standard basis.

Orthonormal frames An orthonormal frame is a pair (x, e) where x ∈ E3 and e = (e1 e2 e3 ) is an orthonormal basis of E3 arranged into the columns of a matrix. e1 x e2

e3

D.1 For any two frames (x, e), (x0 , e0 ), there is a unique rigid motion ϕ, i.e. a distance preserving transformation, taking one to the other: x0 = ϕ(x), e0i = ϕ∗ ei . Each rigid motion ϕ has associated frame (x, e) which is the frame that arises when we apply ϕ to the standard basis frame at the origin. Let O(3) be the set 101

102

The moving frame

of 3 × 3 orthogonal matrices. To each orthonormal frame (x, e) ∈ ⌜E3 , associate the matrix   1 0 h= x e giving a map

h : (x, e) ∈ ⌜E3 7→ h(x, e) ∈ E3 n O(3) .

D.2 Let x0 be the origin and e0 the standard frame, consisting of the standard basis vectors. Associate to each rigid motion ϕ this matrix h(x, e) = hϕ(x0 , e0 ) as above. Prove that this association is an isomorphism of groups from the group of rigid motions of Euclidean space to the group O(3) n E3 of such matrices. We see a matrix description of the rigid motion that takes the origin to x and the standard basis to e. Prove that   1 0 h(x, eg) = h(x, e) 0 g for any orthogonal 3 × 3 matrix g.

Frames on curves Take a curve in E3

and consider all of the frames (x, e) with x a point of the curve.

At each point x of the curve, any two such frames agree up to orthogonal transformation; the set of all such frames is a 4-dimensional manifold. If a rigid motion takes one curve to another, it takes these frames along the one curve to those along the other curve. If the curve is immersed, a frame (x, e) is adapted if e1 is tangent to the curve.

Frames on surfaces

At each point x of the curve, any two adapted frames agree up to changing the direction of e1 , and rotating or reflecting e2 , e3 in the plane normal to the tangent line; the set of all such frames is a surface inside that 4-dimensional manifold. This surface of adapted frames is built by attaching to each point of C four circles: take any one adapted frame, and then rotate e2 , e3 to form one circle, then reflect the plane of e2 , e3 to form another, and then reflect e1 to carry those into two more. We will carry out computations directly inside this surface. For a connected curve, this surface has 4 components. Every rigid motion is determined by where it moves any chosen frame. Any rigid motion preserving an immersed curve takes any frame adapted to the curve to another frame adapted to the curve. So we can guess that the group of rigid motions preserving an immersed curve is at most a 2-dimensional submanifold of the 6-dimensional group of rigid motions. If the curve is a straight line, we can slide a frame along the line, and rotate around the line, so we see a 2-dimensional group of rigid motions.

Frames on surfaces Similarly, a frame is adapted to a surface if x is a point of the surface and e3 is normal to the surface.

The set of adapted frames (x, e) of a surface is a 3-dimensional manifold, as we can move the point x along the surface, and rotate e1 , e2 around e3 . Hence the group of rigid motions preserving an immersed surface is at most a 3-dimensional submanifold of the 6-dimensional group of rigid motions.

The frame bundle The frame bundle ⌜E3 of Euclidean space E3 is the set of all orthonormal frames. Any rigid motion ϕ of E3 takes e1 , e2 , e3 to ϕ∗ e1 , ϕ∗ e2 , ϕ∗ e3 . If you pick an orthonormal frame (x, e) and I pick another one, say (x, e0 ) at the same point x, then yours and mine must agree up to a unique orthogonal matrix, P because they are orthonormal bases of the same vector space E3 . So e0j = i gij ei for a uniquely determined orthogonal 3 × 3 matrix g = (gij ); denote this e0 as eg. An orthonormal frame (x, e) with e = (e1 e2 e3 ) has e ∈ O(3) an orthogonal matrix, so ⌜E3 = E3 × O(3) and the expression eg is matrix multiplication.

103

104

The moving frame

Adapted frames for a curve Take a curve C in E3 . At each point c ∈ C, take e1 to be a unit vector tangent to C. We can pick an adapted frame (x, e), uniquely up to replacing (x, e) by (x, eg) for any orthogonal matrix   ±1 0 g= , h ∈ O(2) . 0 h The curvature vector of C at a point is the acceleration as we move along C at unit speed. The choice of unit speed parameterization is known as an arclength parameterization. The arclength parameterization is unique up to adding a constant or changing the direction of motion, i.e. the sign. Changing direction changes the sign of velocity, but we hit two sign changes when we take two derivatives, so the curvature vector is independent of the choice of arclength parameterization. In particular, the curvature vector is defined without selecting any orientation of the curve C. Write the (perhaps local) inverse of an arclength parameterization as a function s on an open subset of C, uniquely determined up to sign and adding a constant. The identity function x 7→ x on E3 restricts to a function x : C → E3 , and clearly dx , e1 = ds when we take e1 to point in the direction of increasing s, i.e. use s to orient C. Differentiating again, the curvature vector of C is κ ..=

de1 . ds

If κ = 0 at every point of the curve then the curve is a straight line. If κ 6= 0 at some point, then we can uniquely determine a unit normal vector e2 to C by e2 ..=

κ , |κ|

and then define the curvature to be k ..= |κ|. Having chosen e1 , e2 at each point of our curve, e3 is determined up to sign, and can be chosen smoothly. D.3 Calculate that de2 = −ke1 + te3 , ds

de3 = −te2 , dt

for some function t, the torsion. The sign of t depends on the choice of e3 , which is unique up to sign, so te3 is defined independent of choice of e3 . But te3 still depends on the choice of sign of e1 , i.e. the direction of motion. So we could say that the complicated expression te3 ds, a 1-form valued in the normal line to the curve, is an invariant independent of any choices.

The frame e1 , e2 , e3 of an oriented curve of nonzero curvature is the Serret– Frenet frame. Note that we can change sign of e1 , i.e. change orientation, and independently change sign of e3 . So the frame bundle of C contains four Serret–Frenet frames above each point of C. As we move along C, these four points trace out four curves in the frame bundle. The torsion changes sign if we change the sign of e3 , so it is not really a function on the curve C. We can pick out one choice of Serret–Frenet curve by orienting both C and E3 . We can guess that the isometries of E3 which preserve a nonlinear curve are at most 1-dimensional. D.4 For a curve with nonzero curvature vector, prove that its torsion vanishes if and only if the curve lies in a plane. Intuitively, torsion controls the rate at which the curve twists out of its tangent plane, while the curvature controls the rate at which the curve twists in its tangent plane. With only rigid motions, we can turn a circle around, slide a line along itself, and twist a helix along itself, and allow reflections in direction. The adapted frames of these curves are taken one to another by these rigid motions. So the curvature and torsion functions are constant, except that the torsion changes sign if we change the choice of e3 , or of direction of motion along the curve. D.5 Compute the curvature and torsion of a helix. Prove that any constant values of curvature and torsion can arise for a suitable choice of helix. Theorem D.1. Take an interval I ⊂ R. Given a continuous positive function k : I → (0, ∞) and a continuous function t : I → R, there is a curve C in E3 with twice continuously differentiable unit speed parameterization x : I → E3 and with curvature k and torsion t. Any two such curves are identified by a unique rigid motion of E3 . Proof. To each adapted frame (x e), associate the matrix   1 0 g= . x e This g maps each adapted frame (x e) of the curve to the rigid motion g(x e) which takes the origin to x and the standard basis to e. For the adapted frames of a curve, the above gives   0 0 0 0 1 0 −k 0  dg  = g 0 k 0 −t . ds 0 0 t 0

105

106

The moving frame

These equations are linear, so there are global continuously differentiable solutions, by the existence and uniqueness theorem for linear ordinary differential equations, with any initial data. Expand out to dx = e1 , ds 0 k 0  d e1 ds

e2

e3 = e1 

e2

e3



−k 0 t

 0 −t 0

If e1 , e2 , e3 are orthonormal at one point s0 ∈ I, then d e1 · e1 = 2e1 · ke2 = 0, ds etc., so remains an orthonormal frame. The choice of initial conditions at a single point determines the curve and the frame. If we translate x or rotate x, e1 , e2 , e3 by a constant rotation, check that this takes us to another solution. Since e1 (s) is continuously differentiable and dx/ds = e1 , x is twice continuously differentiable.

Any curve of constant positive curvature and constant nonzero torsion is a helix, by existence (from the problem above) and uniqueness. Similarly, any curve of constant positive curvature and zero torsion is a circle, and any curve of zero curvature is a line.

D.6 For a given curve in 3-dimensional Euclidean space, with nowhere vanishing curvature vector, find every other curve so that, parameterized by arc length, points on the two curves which have the same arclength parameter also have the same direction of curvature vector.

Adapted frames for a surface An adapted orthonormal frame (or just frame for short) on a surface S in E3 is an orthonormal frame (x, e) ∈ ⌜E3 so that x ∈ S and e1 , e2 are tangent to S at x, and therefore e3 is normal to S. If S is the unit sphere in E3 , we can write the points of S as   x1 x = x2  x3

107

The soldering forms

with 1 = x21 + x22 + x23 . Away from the north or south pole, we can take     x1 x2 x3  1 1  0  , e2 = p − x21 + x23  e1 = p 2 x1 + x23 −x x21 + x23 x2 x3 1 tangent and

  x1 e3 = x2  x3

normal. Alternatively, we could instead have taken e3 to be the negative or this, and we could also have taken e1 and e2 to be any rotation or reflection of these in the tangent plane. The frame bundle ⌜S is the set of all adapted orthonormal frames for S. The manifold ⌜S is a 3-dimensional submanifold of ⌜E3 = E3 × O(3). D.7 Explain why, if S = S2 ⊂ E3 is the unit sphere around the origin, then ⌜S is diffeomorphic to two disjoint copies of O(3), given by the equation x = ±e3 . Each orthogonal 3 × 3 matrix g preserving the horizontal plane   h 0 g= 0 n with h ∈ O(2) and n ∈ O(1) = {±1}, acts on ⌜S as (x, e)g = (x, eg).

The soldering forms If v is a tangent vector on the manifold ⌜E3 , we can write v as (x, ˙ e), ˙ an infinitesimal motion x˙ of the point x, and an infinitesimal rotation e˙ of the frame e. We can write out any tangent vector x˙ in terms of the basis e1 , e2 , e3 , say as x˙ = a1 e1 + a2 e2 + a3 e3 . The soldering forms are the 1-forms ω1 , ω2 , ω3 on ⌜E3 given by (x, ˙ e) ˙ ωi = ai . So the soldering forms measure, as we move a frame, how the base point of the frame moves, as measured in the frame itself as a basis. The identity function on E3 , which we write as x, is defined as x(p) = p for any point p ∈ E3 . Of course dx(v) = v for any vector v, i.e. dx = I is the identity matrix. We define x also on ⌜E3 by x(p, e) = p. We can write the 1-forms ω1 , ω2 , ω3 on ⌜E3 as ω1 = e1 · dx, ω2 = e2 · dx, ω3 = e3 · dx. On our vector v = (x, ˙ e), ˙ we have v ωi = ei · x˙ = ai .

The connection forms When we move a frame (x, e), the soldering forms measure the motion of the underlying point x. We want to measure the rotation of the vectors e1 , e2 , e3 .

108

The moving frame

Infinitesimal rotations are complicated. Write the inner product on E3 as x · y = x1 y1 + x2 y2 + x3 y3 . So ( 1 if i = j, ei · ej = 0 if i 6= j. If we rotate an orthonormal basis e1 , e2 , e3 through a family of orthonormal bases e1 (t), e2 (t), e3 (t), along some curve x(t), these still have the same constant values of ei (t) · ej (t) at every time t. Differentiate: 0 = e˙ i (t) · ej (t) + ei (t) P· e˙ j (t). Therefore we can write any infinitesimal rotation of frame as e˙ j = i aij ei for an antisymmetric 3 × 3 matrix A = (aij ). The quantity aij measures how quickly ej is moving toward ei . The Levi-Civita connection forms are the 1-forms γij = ei · dej , i.e. v γij = aij : so γij measures the tendency of ej to move toward ei as the frame moves. In particular, 0 = γij + γji . These ωi and γij are defined on ⌜E3 , not on E3 , because they depend on x and e. IfP we move a frame, we said it moves by a velocity vector v = (x, ˙ e) ˙ with e˙ i = j aji ej for an antisymmetric matrix aij , so v γij = v ei · dej = ei · e˙ j = aij . So the 1-forms ωi restrict to any “moving frame” (x(t), e(t)) to describe how the velocity of the moving point x(t) is expressed in the moving frame e(t), while the 1-forms γij describe how the infinitesimal rotation of the moving frame e(t) is expressed at each moment in the moving frame e(t). Write our soldering forms as ω, thought of as a column of 1-forms   ω1 ω = ω2  ω3 and our connection 1-forms as  γ11 γ12 γ = γ21 γ22 γ31 γ32

  γ13 0 γ23  = −γ12 γ33 −γ13

γ12 0 −γ23

 γ13 γ23  0

an antisymmetric matrix of 1-forms, the Levi-Civita connection. D.8 Prove that the soldering and Levi-Civita connection forms satisfying the structure equations of Euclidean space: X dωi = − γij ∧ ωj , j

dγij = −

X

γik ∧ γkj .

k

D.9 For g any 3 × 3 orthogonal matrix, if we write rg (x, e) to mean (x, eg), and ∗ > g> for the transpose of P g, prove that rg∗ ω = g> Pω and rg γ = g γg. Expanding out, this means rg∗ ωi = j gji ωj and rg∗ γij = k` gki γk` g`j .

109

Curves and forms

Curves and forms In this section, we adopt the notation that i, j, k, ` = 2, 3. Take a curve C in E3 . At each point of ⌜C , e2 , e3 are perpendicular to Tx C, so 0 = ω2 = ω3 on ⌜C . Note that ω1 = e1 · dx = ds, if we pick e1 to agree with the direction in which the arclength function s increases. So dω1 = 0. From the structure equations of Euclidean space, 0 = dω2 = −γ21 ∧ ω1 . 0 = dω3 = −γ31 ∧ ω1 . Therefore γ21 = k2 ds, γ31 = k3 ds for unique smooth functions k2 , k3 on ⌜C , but these change sign if we change the direction in which we measure arclength s. We can move any adapted frame (x, e) of C with x moving along C, or with the ei rotating among frames of the tangent line. The 1-forms measuring those motions ω1 , γ23 are all linearly independent, together forming a coframing on ⌜C , i.e. a collection of linearly independent 1-forms spanning the 1-forms at each point. We obtain the structure equations of a curve C in E3 : dω1 = 0, dγ23 = 0. Each of the 4 curves in the frame bundle, from the 4 Serret–Frenet framings, have their own arclength function s, defined to agree in direction with e1 , but only defined up to adding a constant. In other words, since dω1 = 0, locally ω1 = ds for a function s defined up to a constant. On the frame bundle of C, we can turn e2 , e3 , so have a nonzero 1-form γ23 . D.10 Prove that the vector k2 e2 + k3 e3 is actually defined down on C, and equals the curvature vector. If the curvature vector is nonzero, we can define the Serret–Frenet frames to be those on which k2 > 0 and k3 = 0. The set of Serret–Frenet frames consists of 4 curves in the frame bundle. It has k3 = 0, so γ31 = 0, and γ21 = kω1 6= 0, so differentiate 0 = γ31 to get 0 = −γ32 ∧ γ21 , = −kγ32 ∧ ω1 . so γ32 = tω1 for a unique function t on those 4 curves. We can see a general pattern emerging: write out the structure equations, get the geometry to impose some relation on the 1-forms in the structure equations, and differentiate repeatedly, applying Cartan’s lemma when possible, until the invariants pop out, and the structure equations reduce down to having only the linearly independent 1-forms in them.

110

The moving frame

Surfaces and forms The expression dx · dx is the Euclidean inner product in Euclidean space, so is well defined on tangent vectors to any curve or surface in Euclidean space. Take a surface S in E3 . The unit normal e3 is defined up to sign at each point of S. Hence de3 is also defined up to sign, as is de3 · dx, and so e3 (de3 · dx) is defined on S, called the shape operator, or the second fundamental form, denoted II. Let us find a different way to write this. As we move along a curve in S, take e1 to be tangent to that curve, and travel at unit speed; pick e2 , e3 to give an adapted frame along the curve. The vector e2 moves in the direction of e1 , i.e. the curve turns in the tangent plane to the surface, as the ω1 component of e1 · de2 = γ12 . So this component is the curvature of the curve in the surface. Recall that e3 is normal to the surface; stand on the surface, with e3 pointing straight upward. The vector e3 moves down toward e1 , i.e. the surface bends down as we move along the curve, as the ω1 component of e1 · de3 = γ13 . The vector e3 moves toward e2 , i.e. the surface twists like a screw, as we move along the curve, as the ω1 component of e2 · de3 = γ23 . From here on, adopt notation i, j, k, ` = 1, 2. At each point of ⌜S , e3 is perpendicular to Tx S, so ω3 = 0 on ⌜S . From the structure equations of Euclidean space, 0 = dω3 = −

X

γ3i ∧ ωi .

i

P Therefore γ3i = − j aij ωj for unique smooth functions aij = aji on ⌜S . So aij measures the tendency of ei to move toward e3 as we move in direction of ej , i.e. we measure how the surface twists like a screw. This measurement is (not obviously) symmetric in i, j: if the surface twists as we move in a certain direction, it twists just as much if we move in the orthogonal direction. These aij are the components of the shape operator: II = e3 (de3 · dx), = e3 ((ei · de3 )(ei · dx)), = e3 (γi3 ωi ), = e3 (aij ωj ωi ). We can move any adapted frame (x, e) of S with x moving along S and the ei rotating among frames of the tangent space. The 1-forms measuring those motions ω1 , ω2 , γ12 are all linearly independent, together forming a coframing on ⌜S , i.e. a collection of linearly independent 1-forms spanning the 1-forms at each point. On our surface, 0 = ω3 , so often forget ω3 and let ω ..= ω1 + iω2 . To simplify notation, let α ..= γ12 (because the letter α looks like γ rotated in

111

Rotate the frame

the plane). We obtain the structure equations of a surface S in E3 : dω = iα ∧ ω, i dα = Kω ∧ ω ¯ 2 where K is the Gauss curvature. From the structure equations of Euclidean space, dα = dγ12 = −γ13 ∧ γ32 , so K = a11 a22 − a212 .

Rotate the frame This section can be omitted without loss of continuity.

Recall that we write each orthogonal 3 × 3 matrix g preserving the horizontal plane as   h 0 g= 0 n with h ∈ O(2) and n = ±1. Write h as a product of a rotation matrix eiθ and zero or one complex conjugation matrices, which we write as C. Expanding out the equation rg∗ ω = g> ω with ω3 = 0 gives rg∗ ω = h> ω. In complex notation re∗iθ ω = e−iθ ω, ∗ rC ω=ω ¯,

rn∗ ω = ω. Expanding out the equation rg∗ γ = g> γg into 3 × 3 matrices gives

rg∗



rg∗ α = (det h)α,    γ13 γ = nh> 13 . γ23 γ23

If a ..=



a11 a21

 a12 , a22

then rg∗ a = nh> ah. In our complex notation, with α ..= γ12 , re∗iθ α = α, ∗ rC α = −α,

rn∗ α = α.

Mean curvature The mean curvature is H =

1 2

(a11 + a22 ). The mean curvature vector is He3 .

112

The moving frame

D.11 Prove that the mean curvature vector is defined on the surface, a vector field pointing normal to the surface. Why might the mean curvature not be defined as a function on the surface? The sphere E3 of radius r0 has frame bundle consisting of the (x, e) ∈ ⌜E3 so that x = ±r0 e3 . Therefore γi3 = ei · de3 = ±ei · dx/r0 = ±ωi /r0 , so a = ±I/r0 depending on orientation. Gauss curvature is 1/r02 . Mean curvature is H = ±1/r0 , depending on orientation. The mean curvature vector is 2x/r0 , independent of orientation.

For a flat plane, e3 lies perpendicular to the plane, while x, e1 , e2 are tangent to it, so 0 = e3 · dx = e3 · de1 = e3 · de2 , i.e. 0 = ω3 = γ31 = γ32 , so a = 0.

D.12 Compute the shape operator and mean and Gauss curvature of a cylinder of radius r0 . D.13 We know that II is defined on tangent vectors to S, independent of frame; check this again using the rules above for how aij change when we change frame.

Principal curvatures The shape operator of a surface is a symmetric bilinear form on the tangent place Tx S valued in the normal line to the tangent plane, so orthogonally diagonalizable. Its eigenvalues are the principal curvatures. If it has only one principal curvature, i.e. a is a multiple of the identity matrix, the point x is an umbilic of the surface. On the other hand, if it has two distinct principal curvatures then its two perpendicular eigenlines are the principal directions.

Higher fundamental forms D.14 Let Da

..=

2a12 da + α a22 − a11 

 a22 − a11 . −2a12

Differentiate the equations γi3 = aij ωj to reveal that Daij = aijk is symmetric in all lower indices. The third fundamental form is III ..= e3 aijk ωi ωj ωk .

P

k

aijk ωk where

113

Local picture of a surface

Local picture of a surface Take a point x0 of a surface S and let n be a unit normal vector to S at x0 , say n = −e3 at x0 . Fixing that constant value of n, take the linear function f (x) ..= n · x. The differential of f is ei df = ei d (n · x) = n · ei , for i = 1, 2, vanishing at x0 . The second derivative matrix of f at x0 is f 00 (ei , ej ) = ei d (n · ej ) , = ei n · dej , = −ei e3 · dej , = −ei γ3j , = aji . Similarly f 000 (ei , ej , ek ) = aijk . Translate the surface to get x0 the origin, and Tx0 S the horizontal plane, and n the unit vertical vector. Then S is locally the graph of x3 = f (x1 , x2 ), and f has a critical zero at the origin, so x3 =

1 1 aij xi xj + aijk xi xj xk + O(x)4 . 2 6

We can pick the frame e1 , e2 at our point to diagonalize the shape operator, say with eigenvalues (i.e. principal curvatures) k1 , k2 : x3 =

k1 2 k2 2 1 x + x2 + aijk xi xj xk + O(x)4 . 2 1 2 6

D.15 Prove that, if the Gauss curvature is positive at x0 then S lies on one side of its tangent plane near x0 , and if the Gauss curvature is negative then it doesn’t. D.16 Prove that every compact surface in E3 has a point of positive Gauss and mean curvature. Take a smooth function f defined in an open subset of E3 . Let M ⊂ E3 be a level set M ..= (f = c0 ) of f on which df 6= 0. D.17 Compute that on the frame bundle of M , γi3 =

ei D 2 f . |df |

So the shape operator of each level set M = (f = c0 ) of f , at each point x ∈ M where df 6= 0, is D2 f Tx M II = γi3 ωi = . |df |

114

The moving frame

If f is a polynomial function of degree at most two, then D2 f on each tangent plane Tm M of M = (f = c0 ) is just the quadratic terms in f . The nonzero level sets of a positive definite quadratic function have positive mean and Gauss curvature.

Appendix E

Curvature of surfaces

Example: surfaces of revolution Take a curve C in the plane, and rotate it around a line in that plane (the axis of revolution) to create a surface of revolution S. Suppose for simplicity that the curve doesn’t touch the axis of revolution. Place the axis of revolution vertically. Write the points of E3 in cylindrical coordinates (r cos θ, r sin θ, z). Our plane is θ = 0, and our axis of revolution is r = 0. Our plane curve C is arclength parameterized as x0 (s) = (r(s), 0, z(s)). Take adapted frames (x0 , e0 ) for the curve in the plane, but swap e02 and e03 so that e02 is normal to the plane and e03 tangent to the plane but normal to the curve: dx0 = (cos ϕ(s), 0, sin ϕ(s)), ds 0 e2 (s) = (0, 1, 0),

e01 (s) =

e03 (s) = (− sin ϕ(s), 0, cos ϕ(s)). Let ρ(θ) be the rotation matrix rotating E3 about the axis of revolution by an angle θ. Make an adapted frame at each point of the surface as x(s, θ) = ρ(θ)x0 (s) = (r cos θ, r sin θ, z), e(s, θ) = ρ(θ)e0 (s), E.1 Find all ωi , γij . The shape operator: 

γ13 γ23



=



−ϕ˙ 0 sin ϕ 0 − r

  ω1 . ω2

So the Gauss curvature of a surface of revolution is K=

ϕ˙ sin ϕ r¨ =− , r r

while the mean curvature is H=−

1 2

  sin φ ϕ˙ + . r

115

116

Curvature of surfaces

Take a function H(s) and try to construct a curve x0 (s) in the plane, so that the associated surface of revolution has mean curvature H(s). To do this, we have to solve the equations r(s) ˙ = cos ϕ(s), ϕ(s) ˙ = −2H(s) −

sin ϕ(s) r(s)

with r > 0. Each initial value r = r0 > 0 and ϕ = ϕ0 at some s = s0 determines a unique local solution, by the existence and uniqueness theorem for ordinary differential equations. Solve for z by R z(s) = sin ϕ(s) ds, again unique up to adding a constant. E.2 If we want H to be constant, say equal to a constant H0 , can √ you integrate this coupled system of ordinary differential equations? Hint: is r 1 − r˙ 2 +H0 r2 constant along solutions? Similarly, we can prescribe Gauss curvature K(s) arbitrarily by solving the equation r¨ + Kr = 0. For example, the surfaces of revolution with zero Gauss curvature are the circular cones given by rotating a line segment. More generally,pfor constant Gauss curvature K = K0 , the solutions are (with k0 ..= |K0 |)   if K0 > 0, a cos k0 s + b sin k0 s, r(s) = a + bs, if K0 = 0,   a cosh k0 s + b sinh k0 s, if K0 < 0. For K0 > 0, translate s to arrange b = 0, so write r(s) = r0 cos k0 s, so that giving

z˙ 2 = 1 − r˙ 2 = 1 − r02 K0 sin2 k0 s, Z q z= 1 − r02 K0 sin2 k0 s ds,

an elliptic integral, usually denoted E(k0 s | r02 K0 )/k0 . Similar remarks apply to K0 < 0. E.3 Take a curve C0 in the plane R2 . Prove that there is a curve C1 in the plane E2 whose associated surface of revolution satisfies (H, K) ∈ C0 at every point.

117

Orientations

Orientations We could have used only positively oriented frames e1 , e2 , e3 of E3 . We could then look at an oriented surface, and define its adapted frames to have e1 , e2 a positively oriented basis of the tangent plane, and e1 , e2 , e3 a positively oriented basis of E3 . The expression   h 0 g= 0 n simplifies to having h a rotation matrix, and n = 1, so   cos θ − sin θ 0 cos θ 0 , g =  sin θ 0 0 1 which we just write as eiθ . E.4 Over an oriented surface, prove that the 2-form dA ..= ω1 ∧ ω2 is the pullback of a unique 2-form dA on the surface, the area form. E.5 Over an oriented surface, prove that the mean curvature is the pullback of a unique function on the surface. E.6 Prove that, on any surface, oriented or not, the shape operator admits a unique expression as q 2 H q¯ 2 ω + (ω ω ¯ +ω ¯ ω) + ω ¯ . 2 2 2

for functions q : ⌜S → C, H : ⌜S → R. How are the aij related to q, H? The expression qω 2 is the Hopf differential. E.7 If the surface is oriented, prove that the Hopf differential descends to the surface, as a complex valued quadratic form on tangent vectors. The Gauss map of a surface is e3 , mapping the frame bundle to the unit sphere. E.8 Prove that the Gauss map of an oriented surface is the composition of the map (x, e) ∈ ⌜S 7→ x with a unique map x ∈ S 7→ e3 (x), also called the Gauss map.

Recovering a surface from its inner products and shape operator This section can be omitted without loss of continuity.

Theorem E.1 (Bonnet). If two connected oriented surfaces in 3-dimensional Euclidean space are identified by a diffeomorphism preserving orientation and the inner product and shape operator on every tangent space, then the diffeomorphism is induced by an orientation preserving rigid motion of 3-dimensional Euclidean space.

118

Curvature of surfaces

Proof. Note that we use the orientations of the surfaces in identifying the shape operators: we have a well defined normal vector on each surface, so we treat the shape operator as a symmetric bilinear form on tangent spaces. Such a diffeomorphism S → S 0 , clearly identifies positively oriented orthonormal frames e1 , e2 of tangent spaces of S with those of S 0 . Taking the usual orientation of E3 , these extend uniquely into positive oriented orthonormal frames e1 , e2 , e3 = e1 ×e2 , a diffeomorphism of the adapted frames, clearly identifying the soldering 1-forms ω1 , ω2 and ω3 = 0. Differentiate to find that the Levi–Civita connection 1-form γ12 matches. Having the same shape operator ensures precisely that γ31 , γ32 match. It then follows by antisymmetry of γij that all connection 1-forms agree. Let   1 0 h = h(x, e) = x e as above, and h0 in the same way for the corresponding surface:   0 0 h−1 dh = h0−1 dh0 = , ω α under the diffeomorphism. So h and h0 satisfy the same first order linear ordinary differential equation   0 0 dh = h ω α along any curve in the frame bundle ⌜S . Apply a rigid motion to one of the two surfaces, so that their frame bundles contain at least one frame in common. By existence and uniqueness for linear ordinary differential equations, h = h0 along any curve in ⌜S starting at that frame. The adapted frames which are positively oriented for E3 and with e1 , e2 positively oriented on the surface form a connected manifold. Hence h = h0 .

Parallel surfaces This section can be omitted without loss of continuity.

Start with a surface S with a chosen normal direction, i.e. a unit normal vector e3 at each point, smoothly varying. Let S 0 be the translation of S in that direction by a distance t. So then x0 = x + te3 , dx0 = dx + t de3 and e3 ·dx0 = e3 ·dx+te3 ·de3 = 0, so e3 is still normal to S 0 at x0 . Hence Tx0 S 0 = Tx S ∼ ⌜S by is constant in t, and the Gauss map is unchanged. Identify ⌜S 0 = Φ : (x, e) 7→ (x0 , e0 ) = (x + te3 , e). The soldering 1-forms are ωi0 = ei · dx0 , = ei · dx + tei · de3 , = ωi + tγi3 , = ωi + taij ωj ,

119

Constant mean curvature surfaces

which we can write as ω 0 = (I + ta)ω. The Levi-Civita connection 1-form is 0 γ12 = e1 · de2 = γ12 .

The shape operator is 0 γi3 = ei · de3 ,

= γi3 , = aij ωj 0 If we write this as γ∗3 = aω, then 0 γ∗3 = aω = a(I + ta)−1 ω 0 ,

so

a0 = a(I + ta)−1 .

In any frame in which a is diagonalized, so is a0 :   k1 0 a= 0 k2 in principal curvatures, so  0 k1 a0 = 0

0 k20



=

k1 1+tk1

0

0

k2 1+tk2

! .

Hence Gauss curvature K0 =

det a K = det(I + ta) 1 + 2Ht + Kt2

and mean curvature

H + Kt . 1 + 2Ht + Kt2 In particular, the parallel surface will not be smooth enough to compute such quantities at any point x0 = x + te3 where 1 + tk1 or 1 + tk2 vanish. But if these don’t vanish, then the differential of the map is H0 =

dx0 = dx + t de3 = ωi ei + taij ωj ei an immersion. Surfaces parallel to a Weingarten surface are also Weingarten, but for a different relation between mean and Gauss curvature.

Constant mean curvature surfaces This section can be omitted without loss of continuity.

On an oriented surface, a complex differential is an expression of the form f ω k on the frame bundle, but which is in fact defined on the surface. In other words, it doesn’t transform as we right translate by eiθ . Since ω transforms by e−iθ , we need f to transform by eikθ : re∗iθ f = eikθ f.

120

Curvature of surfaces

E.9 Differentiating this, show that ¯ ω df + ikf α = Df ω + Df ¯, ¯ . for unique complex functions Df, Df ¯ , i.e. df +ikf α is complex A holomorphic differential is one for which 0 = Df linear. A holomorphic chart on an oriented surface in E3 is a chart on an open subset of the surface with differential dz a holomorphic differential. Any two such charts locally agree up to complex analytic transition maps. The Korn– Lichtenstein theorem says that any oriented surface is covered by domains of holomorphic charts. Hence any oriented surface in E3 is a Riemann surface. E.10 Compute that the Hopf differential satisfies ¯ ω dq + 2iqα = Dqω + Dq ¯, ¯ = 2(H1 − H2 i) where H is the mean curvature, and dH = H1 ω1 + where Dq H2 ω2 . Thereby, prove: Theorem E.2. An oriented surface has constant mean curvature just when its Hopf differential is holomorphic. The Riemann–Roch theorem [13] p. 99 shows that a compact Riemann surface of genus g has a space of quadratic differentials of complex dimension:   g = 0, 0, 1, g = 1,   3g − 3, g ≥ 2. It is well known that there is precisely one genus zero compact Riemann surface, up to holomorphic diffeomorphism: the Riemann sphere [13] p. 125. Applying Bonnet’s theorem: Corollary E.3. For any constant H0 , the sphere has at most one immersion in E3 with constant mean curvature equal to H0 , up to rigid motion: as a round sphere of radius 1/H0 .

Away from umbilics Recall that principal curvatures of a surface are the eigenvalues of its shape operator. Suppose that they are distinct k1 6= k2 , i.e. there are no umbilic points. A Darboux frame is one in which e1 , e2 are eigenvectors of the shape operator with eigenvalues k1 < k2 . The set of Darboux frames of the surface is a surface, as each Darboux frame is uniquely determined up to signs. On that surface γi3 = ki ωi , for i = 1, 2.

121

Away from umbilics

E.11 Differentiate the equations of a Darboux frame to see that      k1 k11 k12 ω1 d = , k2 k21 k22 ω2 for some functions kij , perhaps not symmetric in lower indices, while γ12 =

k12 ω1 + k21 ω2 . k2 − k1

Expand 0 = d2 ki to find that, if we let Dki1 = dki1 + ki2 γ12 , Dki2 = dki2 − ki1 γ12 , then Dkij = kijk ωk with kijk = kikj . Recall that Gauss curvature is K = k1 k2 . Plug into the equation dγ12 = Kω1 ∧ ω2 . to get  d

k12 ω1 + k21 ω2 k2 − k1



= k1 k2 ω1 ∧ ω2 .

Expanding this out gives 0=

2 k122 − k211 2k 2 + 2k21 + k1 k2 . + 12 k2 − k1 (k2 − k1 )2

Lemma E.4. If a surface bears a point at which the larger principal curvature is maximal and the smaller minimal, then either the point is umbilic or the Gauss curvature is not positive at that point. Proof. We can assume that k2 > k1 near that point, and then adapt frames as above. At a critical point of k2 and k1 , 0 = dk1 = dk2 , so kij = 0 and so γ12 = 0, so Dkij = dkij = kijk ωk . Since k2 is maximal, along the flow of e1 , k211 ≤ 0. Since k1 is minimal, along the flow of e2 , k122 ≥ 0. Plug in above to get K = k1 k2 ≤ 0. Theorem E.5 (Liebmann). Suppose that S is a connected compact surface in E3 of (i) constant Gauss curvature or (ii) constant mean curvature with Gauss curvature not having any critical zeroes. Then S is a round sphere. Proof. By problem D.16 on page 113, there is a point of positive Gauss and mean curvature, so the constant is a positive constant. At any point at which the principal curvatures are as far apart from one another as possible, one is maximal and the other minimal, as their product is constant. Apply lemma E.4 to find a point of nonpositive Gauss curvature where the Gauss curvature is critical.

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Curvature of surfaces

E.12 Suppose that S is a connected surface in E3 and that one principal curvature is a nonzero constant, and always distinct from the other. Prove that S is a union of circular arcs of constant radius, with centres on a curve. We define completeness in appendix G; for the moment, we only need know that on a complete surface, every bounded vector field is complete. Theorem E.6 (Hilbert). No complete surface in E3 has constant negative Gauss curvature. Proof. Take a surface of constant negative Gauss curvature; rescale to make the Gauss curvature equal to −1. We can assume the surface is connected, and replace it by its universal covering space if needed, so assume it is simply connected. Let u be half the angle between the asymptotic curves, so 0 < u < π/2. e2 u e1

A surface of negative Gauss curvature has no umbilics. Check that the principal curvatures are − cot u, tan u. Differentiate the equations γ13 = − cot u ω1 , γ23 = tan u ω2 . Differentiate both of these equations, using the structure equations, to find γ12 ∧ ω1 = − tan u du ∧ ω2 , γ12 ∧ ω2 = − cot u du ∧ ω1 . Check that ω1 / sin u and ω2 / cos u are closed 1-forms; noting that 0 < u < π/2, so the denominators don’t vanish. Because the surface is connected and simply connected, there are functions s1 , s2 so that ds1 = ω1 / sin u, ds2 = ω2 / cos u, unique up to constant and with ds1 , ds2 linearly independent. R The surface is complete, so e1 , e2 have complete flows. The indefinite integral ω1 along curves ω2 = 0 is an unbounded function, defined up to constant. But s1 grows at least as fast, so is also unbounded along curves on which ds2 = 0, and vice versa. So the map (s1 , s2 ) from our surface to the plane is onto, and a local diffeomorphism. The vector fields ∂s1 = sin u e1 , ∂s2 = cos u e2 are complete and commuting. Flows of these for times s1 , s2 invert our previous map, hence a diffeomorphism.

123

Away from umbilics

Let x ..= (s2 − s1 )/2, y ..= (s1 + s2 )/2. The level sets of x and y are precisely the asymptotic curves. Check that γ12 = −ux dx + uy dy, and that the relation dγ12 = Kω1 ∧ ω2 becomes the sine–Gordon equation for the angle θ = 2u: θxy = sin θ. Note also that 0 < θ < π, so sin θ > 0. Apply Stokes’s theorem to a rectangle R in the xy-plane, using the xycoordinate orientation on R: Z Z γ12 = dγ12 , ∂R

R

to find θ(x, y) − θ(0, y) = θ(x, 0) − θ(0, 0) +

Z

sin θ dx dy.

Two observations from this equation: • The difference in values of θ along the top of any rectangle is at least as large as the difference along the bottom since sin θ > 0. • The area of R is the integral of ω1 ∧ ω2 , in the orientation of the surface. In the xy-plane orientation, it is Z Z − ω1 ∧ ω2 = dγ12 , R ZR = sin θ dx dy, = θ(x, y) − θ(0, y) − θ(x, 0) + θ(0, 0). Since 0 < θ < π, the area of the rectangle, in the geometry of our surface, is less than 2π. Our coordinates are global, so any measureable subset of the surface has area no more than 2π. Let us compare to the area as measured in the Euclidean metric of the xy-plane. Each set of infinite Euclidean area has only at most 2π area in the surface geometry, i.e. at most 2π integral of sin θ. So that set contains sets of large Euclidean area where sin θ gets small, so θ gets close to 0 or π. Suppose that θ is not constant on some horizontal line. Change the sign of x and y if needed, to arrange that θ increases along a horizontal line segment. Split that line segment into three line segments. On the leftmost, suppose that θ increases by some amount εL , and on the rightmost by some amount εR .

εL

εR

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Curvature of surfaces

Draw the half strips above the line segments. So θ increases by at least εL across each parallel line segment across the left half strip. But θ > 0 everywhere, so in the middle strip, θ > εL . Similarly, in the middle strip, θ < π − εR . So sin θ is bounded away from zero in the middle strip, a contradiction to the infinite Euclidean area of the strip. Hence θ is constant on every horizontal line, i.e. 0 = θx , so 0 = θxy = sin θ, a contradiction. E.13 Use the Frobenius theorem to construct a surface of constant negative Gauss curvature out of any smooth solution of the sine–Gordon equation, reversing the story in Hilbert’s theorem.

Appendix F

The Gauss–Bonnet theorem

We prove that the integral of Gauss curvature over a compact oriented surface in 3-dimensional Euclidean space is a diffeomorphism invariant.

Length and curvature of curves F.1 Suppose that x(t) is a smooth curve in E3 so that x0 (t) 6= 0 for every t with t0 ≤ t ≤ t1 . Prove that there is a reparameterization y(t) = x(τ (t)) for some function τ (t) so that y 0 (t) is a unit length vector, i.e. y(t) has unit speed. Prove that τ (t) is uniquely determined up to adding a constant. Prove that the length of y(t) for a ≤ t ≤ b is equal to the length of x(t) for τ (a) ≤ t ≤ τ (b). Take an oriented curve C, in other words an oriented 1-dimensional manifold, smoothly immersed in a surface S. An adapted frame for C at a point c ∈ C is a pair (c, e) for S, where (x, e) ∈ ⌜S and x is the point of S at which the point c ∈ C lies, and e1 is a positively oriented basis of the tangent line to C at m. Let ⌜SC be the set of all adapted frames for C. F.2 Prove that the set ⌜SC is a curve in C × ⌜S . Map (c, e) ∈ ⌜SC 7→ (x, e) ∈ ⌜S . Using this map to pullback ω1 , ω2 and α = γ12 . If we parameterize C, say as x(t), then e1 (t) is the unit vector in the direction of x0 (t), so that ω1 = e1 (t) · dx = e1R(t) · dx ˙ dt. So dt dt = |x(s)| ω1 is a well defined 1-form on the curve C and C ω1 is the length of the curve C. Meanwhile e2 is perpendicular to the tangent line to C, so on ⌜SC , ω2 = e2 · dx = e2 · dx dt dt = 0. Therefore 0 = dω2 = α ∧ ω1 . Applying Cartan’s lemma, α = −κ2 ω1 for some function κ2 . The 1-form ω1 is a basis for the 1-forms on ⌜SC . The equations ω2 = 0 and α = −κ2 ω1 are precisely the linear equations that cut out the tangent spaces of ⌜SC inside ⌜S . F.3 Prove that κ2 e2 , the geodesic curvature of the curve C at the point c, is the projection of the curvature vector of C (as a space curve in E3 ) to the tangent plane of S. F.4 Prove that the geodesic curvature of an oriented curve is unchanged if we reorient the curve.

125

126

The Gauss–Bonnet theorem

A flat plane P ⊂ E3 is perpendicular to a constant unit vector e3 , so 0 = de3 = e3 · dx = e3 · de1 = e3 · de2 . Therefore on ⌜P , the shape operator vanishes. Given a curve in the plane, the adapted frames have ω2 = 0 and α = −κ2 ω1 . If the curve is given by s 7→ (x1 (s), x2 (s)) in an arclength parameterization then we write the velocity vector as a e1 = (x˙ 1 , x˙ 2 ) = (cos φ, sin φ) , and then, rotating by a right angle, e2 = (−x˙ 2 , x˙ 1 ) = (− sin φ, cos φ) , so that

˙ ds, −α = e2 · de1 = φ(s)

˙ i.e. κ = φ(s), the curvature is the rate of rotation of the unit tangent vector. We can recover a curve from its curvature κ, by integrating κ to find φ, and then integrating dx = (cos φ, sin φ) ds to recover the curve. Constants of integration represent rotation and translation of the curve. To get a curve of constant curvature κ = κ0 in the plane, we integrate to find φ = κs, up to a constant, and then up to a constant, x(s) =

1 (− sin κ0 s, cos κ0 s) , κ0

a circle. Lemma F.1. Take a curve C on a surface S in E3 . Suppose that C has curvature vector as a space curve nowhere perpendicular to S. Take θ to be the angle between that curvature vector and its projection to the surface S. Its surface invariants (geodesic curvature vector κ and shape operator a) are related to its invariants as a space curve (the curvature k and torsion t of C) by κ = k cos θ e2 geodesic curvature a(x, ˙ x) ˙ = k sin θ e3 normal curvature a(x, ˙ x˙ ⊥ ) = (θ˙ + t) e3 geodesic torsion where x˙ is its unit speed velocity in either orientation, and x˙ ⊥ is the unit tangent vector to S normal to C which is closest to the curvature vector. Proof. Write the Serret–Frenet frame to C as e1 , e02 , e03 , and all adapted frames to the surface S for which e1 is tangent to the curve C as e1 , e2 , e3 . Since

127

The Gauss map

e1 is the same for both frames, the other two pairs of legs are orthogonal transformations of one another: e02 = cos θe2 ∓ sin θe3 , e03 = sin θe2 ± cos θe3 . The angle θ is at most π/2, and equal just when the curvature vector as a space curve is perpendicular to the normal line on S, which we suppose never happens. Pick e3 so that ± = +. The Serret–Frenet equations are x˙ = e1 , e˙ 1 = ke02 , e˙0 2 = −ke1 + te03 , e˙0 3 = −te02 . Geodesic curvature is κds = −α. The connection forms are     −α γ21 −a11 ds = γ31  , −a12 ds γ32   e2 e˙ 1 = e3 e˙ 1  ds, e3 e˙ 2   k cos θ  ds. ±k sin θ = d 0 0 e3 ds (cos θe2 + sin θe3 ) F.5 Expand out the last line to finish the proof.

F.6 Suppose that a surface has a straight line on it, through every point, in every tangent direction through that point. Prove that it is a plane.

The Gauss map Consider first a surface S in E3 . On its frame bundle we have ω1 , ω2 and γ12 , so that dω1 = −γ12 ∧ ω2 , dω2 = γ12 ∧ ω1 , dγ12 = Kω1 ∧ ω2 . where K is the Gauss curvature of the surface. We can find a geometric description of Gauss curvature. Take the unit vector e3 , which is uniquely

128

The Gauss–Bonnet theorem

determined at each point of S up to sign, being normal to S. Differentiate this vector, and write its derivative in the frame itself: de3 = (e1 · de3 ) e1 + (e2 · de3 ) e2 + (e3 · de3 ) e3 , = γ13 e1 + γ23 e2 + γ33 e3 , but γ33 = 0 because γ is antisymmetric, = (a11 ω1 + a12 ω2 ) e1 + (a12 ω1 + a22 ω2 ) e2 . So if we move in the e1 direction, the unit vector e3 moves by a11 e1 + a12 e2 , while if we move in e2 direction, the unit vector e3 moves by a12 e1 + a22 e2 . The unit square e1 ∧ e2 in the tangent plane Tx S maps by e03 = de3 to the parallelogram  e3∗ (e1 ∧ e2 ) = a11 a22 − a212 e1 ∧ e2 . But the Gauss curvature is K = a11 a22 − a212 . Therefore the Gauss curvature is the area stretch factor of the Gauss map e3 . While e3 is only defined on the frame bundle of S, nonetheless it is defined locally on S up to sign; the sign doesn’t alter the value of the area stretch factor. Gauss curvature also doesn’t depend on orientation.

The Gauss–Bonnet theorem Take a vector field X on an oriented surface S. Away from the zeroes of X, the unit vector field X e1 ..= |X| is defined. Extend it to an oriented orthonormal basis e1 , e2 . This choice of orthonormal frame gives a section of ⌜S → S over the points where X 6= 0. Take a point x0 ∈ S at which X has an isolated zero. In any positively oriented chart near x0 , draw a circle C0 around x0 , so that x0 is the only zero of X inside the circle. The index of X at x0 is the number of times that the frame e1 , e2 turns around the origin, as measured in some choice of orthonormal frame e01 , e02 defined near x0 . Write     0  e1 cos θ − sin θ e1 = e2 sin θ cos θ e02 near C0 . The angle θ varies by a multiple of 2π, say 2πn0 , around C0 , and the index of X at x0 is n0 . Clearly the index doesn’t change if we continuously vary X while the surface S moves continuously in space E3 . The sum of indices is the Euler characteristic χX . F.7 Prove that the vector field X = ax

∂ ∂ + by ∂x ∂y

on the plane has index 1 if a, b > 0 or a, b < 0 and index −1 if a and b have opposite signs.

129

Degree

F.8 Calculate out that α = e2 · de1 = e02 · de01 − dθ = α0 − dθ. The Euler characteristic of a compact oriented surface S is Z 1 χS ..= K dA. 2π S Theorem F.2. Suppose that X is a vector field on a compact oriented surface S with only isolated zeroes. Then χX = χS . Proof. Take a set Dk diffeomorphic to a closed disk in some chart around each zero xk of X and let Ck = ∂Dk . The framing e1 , e2 lifts [ S 0 ..= S − Dk k

to a compact surface with boundary in ⌜S . Integrate: Z Z K dA = dα, 0 S0 ZS = α, ∂S 0 XZ =− α, k

=−

k

=−

so that

Z S

e01 · de02 +

XZ

Ck

XZ k

e1 · de2 ,

Ck

XZ k

=−

Ck

XZ

k

K dA +

Ck

K dA = 2π

X

dθ,

Ck

2π indexxk X,

k

X

indexxk X.

k

Degree Suppose that ϕ : P → Q is a continuously differentiable map between two oriented manifolds of equal dimension, perhaps with corners. The sign of ϕ at a regular point p ∈ P is 1 if ϕ is orientation preserving near p, and −1 is ϕ is orientation reversing near p. Suppose that q0 ∈ Q is a regular value and that

130

The Gauss–Bonnet theorem

there are only finitely points p ∈ P mapped to q0 by ϕ. The degree of ϕ above q0 is the sum of signs at these points p: X degq0 ϕ = (sign of ϕ at p) . ϕ(p)=q0

The following two theorems are proven in [27]. Theorem F.3. Consider a proper continuously differentiable map ϕ : P → Q between oriented manifolds (perhaps with corners). Suppose that Q is connected. The degree of ϕ is the same at all regular values. Theorem F.4. Suppose that ϕ : P → Q is a proper continuously differentiable map between two oriented manifolds (perhaps with corners) of the same dimension, say n, and that Q is connected. For any n-form ω on Q, which vanishes outside some compact set, Z Z ϕ∗ ω = deg ϕ ω. P

Q

The Gauss map For any oriented surface S in E3 , the Gauss map e3 taking each point x ∈ S its unit normal vector, a point of the unit sphere, has degree R K dA χS deg e3 = = . 4π 2 Theorem F.5. On any compact oriented surface S in E3 with Gauss map e3 : S → S2 , 2 deg e3 = χX , for any vector field X on S with only isolated zeroes. Theorem F.6. Every compact oriented surface S in E3 , twice continuously differentiable, has a continuously differentiable vector field X with only finitely many zeroes, each of index 1 or −1. Proof. The Gauss map is onto: take any unit vector n, and look at the points x ∈ S on which the linear function f (x) ..= hn, xi has its maximum value. By Sard’s theorem, almost every point n in the unit sphere is a regular value of the Gauss map. For such a value of n, the differential of the Gauss map has full rank. Its differential has determinant given by the Gauss curvature, so the Gauss curvature is nonzero at every point x ∈ S for which ν(x) = n. The function f restricted to S has critical points at the points x where n is perpendicular to Tx S. Again by Sard’s theorem, we can arrange that these points are regular values of the Gauss map, i.e. have nonzero Gauss curvature. In any orthonormal framing e1 , e2 , e3 , the differential of f is ei df = ei (n · x) = n · ei ,

131

The Gauss map

for i = 1, 2. The second derivative matrix of f at a critical point is f 00 (ei , ej ) = ei d (n · ej ) , = ei n · dej , = ei (±e3 ) · dej , = ±ei γ3j , = ∓aji . So the function f has f 00 a rank 2 quadratic form on each tangent space where f is critical. In particular, if we write df = f1 ω1 + f2 ω2 , then the vector field X = ∇f = f1 e1 + f2 e2 has only finitely many zeroes. Since we can vary our surface continuously without changing indices, we can see that the index doesn’t change as we vary S and f so that f remains critical at some point x0 with f 00 nondegenerate at x0 . In particular, we can arrange that S is flat near x0 with f a quadratic function instead, without changing the index. But we saw above that the index of X = ∇f at x0 is then 1 if det f 00 (x0 ) > 0 and −1 if det f 00 (x0 ) < 0. Corollary F.7. Diffeomorphic surfaces have equal Euler characteristic.

Appendix G

Geodesics on surfaces

Geodesics are locally shortest paths on a surface. We prove that they exist and that they are smooth.

Geodesic flow Each point (xo , eo ) ∈ ⌜E3 moves along a path x(t) = xo +teo1 , e(t) = eo . In other words, don’t rotate the frame, but slide the base point x along the direction of the first leg of the frame. ...

...

Define a vector field X(x, e) ..= (e1 , 0) on ⌜E3 ; these paths are its flow lines. Imitate this vector field on the frame bundle of a surface S in E3 . Write points of ⌜S × E2 as (x, e, y) for y ∈ E2 . The geodesic spray of S is the vector field X on ⌜S × E2 given by X (ω, α, dy) = (y, 0, 0) This expression does not involve γ13 or γ23 , since these are not linearly independent 1-forms on ⌜S . The flow of X is the geodesic flow. P The vector field X moves every frame (x, e), moving x in the direction of yi ei , and “not rotating” the frame e, since X γ12 = 0. Intuitively, X twists frames only in the normal directions (in the γ13 , γ23 directions), not tangentially (the α = γ12 direction), while keeping e1 , e2 tangent to the surface S. Map (x, e) ∈ ⌜S → (x, e, (1, 0)) ∈ ⌜S × E2 , so that ⌜S is identified with a submanifold of ⌜S × E2 , making ⌜S invariant under the geodesic flow. Write a> for the transpose of a matrix a. Let g ∈ O(2) × ±1 act on ⌜S × E2 by (x, e, y)g = (x, eg, a> y). if g=

 a 0

 0 . b

 Lemma G.1. On ⌜S × E2 , rg∗ (ω, γ12 , dy) = g> ω, γ12 , g> dy while rg∗ X = X. 133

134

Geodesics on surfaces

Proof. The reader can check how the 1-forms pull back. The equations defining X imply invariance under O(2) × ±1. P Let p : (x, e, y) ∈ ⌜S × E2 7→ (x, v) ∈ T S by v = yi ei . Proposition G.2. There is a uniquely determined vector field X on T S, also called the geodesic spray, so that p∗ X = X. Proof. Calculate that at a point z = (x, e, y), for any g ∈ O(2) × ±1, p0 (zg)X(zg) = p0 (zg)rg0 (z)X(z), 0

= (p ◦ rg ) (z)X(z), = p0 (z)X(z). The reader can check that O(2) × ±1 acts transitively on the points z = (x, e, y) lying above a given point (x, v) = p(z). Therefore the vector p0 (z)X(z) is independent of the point z, depending only on (x, v) = p(z). A geodesic is a curve in S which is a projection to S of a flow line of X in T S. G.1 Take a surface S in E3 . Prove that any geodesic x(t) of S, as a curve in E3 , has geodesic curvature a(x, ˙ x). ˙ Prove that a = 0 just when all geodesics of S are geodesics of E3 , i.e. straight lines, so that S is locally an open subset of a plane. Lemma G.3. A smooth curve C on a surface S in E3 has vanishing geodesic curvature just when it is a geodesic. Proof. The geodesic spray vector field X points along the tangent line cut out by the equations ω2 = 0 and α = 0 and has X ω1 = 1. The tangent spaces of ⌜SC are cut out by the equations ω2 = 0 and α = −κ2 ω1 . The geodesic spray X lies tangent to ⌜SC just when the geodesic curvature of C vanishes, which therefore occurs just when the projection of the flow lines of X lie along C, i.e. C is a geodesic.

First order deformation Take a plane curve (x(s), y(s)) ∈ R2 . Draw a vector field along this curve: a vector (u(s), v(s)) at each point (x(s), y(s)). The smooth homotopy s, t 7→ (x(s) + tu(s), y(s) + tv(s)) pushes out each point (x(s), y(s)) in the direction of our vector field. In the same way, if we have any compactly supported vector field v(s) along a curve x(s) in an open set in E3 , we can let xt (s) = x(s) + tv(s), and then xt (s) will be defined for some interval of time t. Alternatively, we could create many other smooth homotopies xt (s) with x0 (s) = x(s) and ∂ ∂t t=0 xt (s) = v(s); we say that v(s) is the first order deformation of xt (s). A vector field v along a curve x(s) on a surface S is a smooth choice of vector v(s) ∈ Tx(s) S at each point. As long as the vector field v(s) has compact

Smooth homotopies and variation of length

support in a single chart, we can use the chart to identify an open set of the surface with an open set in the plane, the curve with a plane curve, and each vector field along the curve with a vector field along a plane curve, so create a homotopy with given first order deformation. Lemma G.4. Take a curve C (perhaps with boundary) on a surface S in E3 , and a compactly supported vector field v along C. Then v is a sum v = v1 + v2 + · · · + vk of vector fields along C, so that each vector field vj is the first order deformation of a smooth homotopy of C. Proof. Cover C by charts and take a partition of unity.

Smooth homotopies and variation of length Lemma G.5. On a surface S in E3 , take a vector field v(s) along a unit-speed curve x(s) on S. Suppose that v vanishes at s = s0 and at s = s1 . Let κ(s) be the geodesic curvature of x(s). Suppose that there is a smooth homotopy xt (s) with first order deformation v(s) along x0 (s) = x(s), so xt (s0 ) = x (s0 ) and xt (s1 ) = x (s1 ) for all values of t. Then Z d length x = κ · v ds t dt t=0 Proof. Our homotopy is a map x : X → S for some domain X ⊂ R2 in the s, tplane. Since x(s) has nowhere zero velocity, by continuity the curve s 7→ xt (s) has nonzero velocity for t near 0. If we restrict t to a small enough interval, we can ensure that ∂x ∂s 6= 0 for all (s, t). An adapted frame for the homotopy x(s, t) is a frame e so that ∂x ∂s is a positive multiple of e1 , and therefore is perpendicular to e2 , e3 . So e1 = e1 (s, t) is a field of vectors, pointing in the direction of each curve in the homotopy. Since e1 = e1 (s, t) is actually now defined on the st-plane, ω1 = e1 dx becomes well defined as a form on the st-plane as well. At each point (s, t, e) of the adapted frame bundle, we can ∂x write the first deformation v = ∂x ∂t of each of the curves xt as ∂t = v1 e1 + v2 e2 . Denote the set of adapted frames of the homotopy by ⌜Sx ; we leave the reader to check that this set is a manifold. On ⌜Sx , we have ω2 = v2 dt. Taking exterior derivative of both sides of this equation, 0 = dω2 − dv2 ∧ dt, = α ∧ ω1 − dv2 ∧ dt. By Cartan’s lemma (lemma 1.2 on page 6), there are unique smooth functions κ2 , a2 , b2 on ⌜Sx so that −α = κ2 ω1 + a2 dt, dv2 = a2 ω1 + b2 dt. Checking dimensions, the tangent spaces of ⌜Sx are cut out by the equations ω2 = v2 dt and −α = κ2 ω1 + a2 dt.

135

136

Geodesics on surfaces

Let `(t) = length xt and let π(s, t) = t. Write π∗ for integration over the fibers of π. Z `(t) = |x˙ t | ds, Z = x∗ ω1 , = π∗ x∗ ω1 . Therefore d` = π∗ x∗ dω1 , = π∗ x∗ (−α ∧ ω2 ) , X = π∗ x∗ (κ2 ω1 ∧ v2 dt) , i

= π∗ x∗ (κ · vω1 ∧ dt) , Z  = κ · vω1 dt. Along t = 0, since our curve is unit-speed, ω1 = ds. Lemma G.6. If a smooth curve locally minimises length between its points, then it is a geodesic. Proof. R If C is a smooth curve of minimal length between two points, then 0 = κ · v ds for any first order deformation v. Any R vector field v along C is a sum of first order deformations, so again 0 = κ · v ds. In particular, we can take v a positive multiple of κ near some point of C, and zero away from there.

The exponential map Write the flow of X on the tangent bundle as (x, v) 7→ etX (x, v), defined for all (x, v) with v close enough to zero. The projection map is πS : (x, v) ∈ T S 7→ x ∈ S. For each point x ∈ S, the exponential map is expx : v ∈ open ⊂ Tx S 7→  πS eX (x, v) ∈ S. The exponential map might only be defined near v = 0. G.2 Prove that πS0 (x, v)X(x, v) = v. Lemma G.7. The exponential map on a surface S in E3 has derivative exp0 (0) = id : T0 Tx S = Tx S → Tx S. Proof. Going back to the definition of X, for any constant c ∈ R, cX(x, e, y) = X(x, e, cy). Therefore the flow satisfies etX (x, e, cy) = ectX (x, e, y) for any con-

137

The exponential map

stant c ∈ R. In particular, d X(x, e, y) = etX (x, e, y), dt t=0 d eX (x, e, ty). = dt t=0 Taking the projection down to T S, d πS ◦ eX (x, tv). v= dt t=0 This gives us exp0x (0)v

d = expx (tv), dt t=0 d = πS ◦ eX (x, tv), dt t=0 d = πS ◦ etX (x, v), dt t=0 = πS0 (x, v)X(x, v), = v.

We need locally uniform control on geodesics. The injectivity radius of a surface S in E3 at a point x ∈ S is the supremum of numbers r > 0 so that expx is a diffeomorphism on the ball of radius r in Tx S. Lemma G.8. For any compact subset K ⊂ S of a surface S in E3 there is a lower bound r = rK > 0 on the injectivity radius of all points x ∈ K. If x ∈ K and y ∈ S is within distance r of x then there is a unique geodesic of length less than r from x to y. Proof. Consider the map f : T S → S × S given by f (x, v) = (x, expx v). As above, f 0 (x, 0)(u, v) = (u, u + v), so f is a diffeomorphism of an open neighborhood of UT S ⊂ T S of { (x, 0) | x ∈ S } to a open neighborhood US×S of { (x, x) | x ∈ S } ⊂ S × S. Take any compact set K ⊂ S. Consider a sequence of points xj , yj with xj ∈ K and yj ∈ S with distances between xj and yj getting small. By compactness, after replacing with a subsequence, our sequence converges to a point (x, x), which lies in US×S . But US×S is open, so all but finitely many pairs (xj , yj ) belong to US×S . Suppose that there are arbitrarily close points xj , yj with xj ∈ K with no unique shortest geodesic. Then (xj , yj ) is not in the image of f , i.e. not in US×S , a contradiction. Similarly, we cannot find arbitrarily close points xj , yj with xj ∈ K where f −1 is not defined. But f −1 (x, y) = exp−1 x y, so expx y is defined for near enough points x and y, and is a diffeomorphism because f is a diffeomorphism.

138

Geodesics on surfaces

Geodesic polar coordinates G.3 Suppose that η1 , η2 are an orthonormal coframing of an oriented surface S in E3 , i.e. η1 , η2 yield an orthonormal basis of each cotangent space. Let u1 , u2 be the dual vector fields, also orthonormal, and take u3 to be the positively oriented normal vector field. Define a map f : x ∈ S 7→ (x, u1 (x), u2 (x), u3 (x)) ∈ ⌜S . Prove that f ∗ ωi = ηi . For this reason, we prefer to just denote any choice of orthonormal coframing η1 , η2 as ω1 , ω2 , and the associated 1-form f ∗ γ12 as just γ12 , which we say is the connection 1-form of this coframing. Take the exponential map expx0 and use it to identity Tx0 S with an open set around x0 in S. Make an isometric linear isomorphism Tx0 S ∼ = E2 . Take polar coordinates on E2 . The direction ∂r is, by definition of the exponential map, the direction of the geodesics moving at unit speed away from x0 , so is a unit vector field. Take ω1 , ω2 an orthonormal coframing so that ω1 = 1 and ω2 = 0 on ∂r . So ω1 = dr + a(r, θ) dθ, ω2 = h(r, θ) dθ. Differentiate, denoting

∂h ∂r

as hr , and so on, to find that α=−

ar ω1 + hr ω2 . h

The curves ω2 = 0 have geodesic curvature ar /h, which vanishes because they are geodesics, so a = a(θ). G.4∗ Prove that a(θ) = 0 and h(r, θ) → 0 and hr (0, θ) → 1 as r → 0. Differentiate α to find the Gauss curvature K. Summing up: Lemma G.9. Any surface, in geodesic normal coordinates, has orthonormal coframing, connection form and curvature: ω = dr + ih(r, θ)dθ, α = −hr dθ, hrr K=− , h h(r, θ) → 0 as r → 0, hr (0, θ) → 1 as r → 0, h(r, θ + 2π) = h(r, θ). The equation hrr + Kh = 0 is linear in h, and uniquely determines h subject only to h = 0 and hr = 1 at r = 0. By uniqueness h(r, θ + 2π) = h(r, θ).

139

Geodesics are locally minimal

For example, if K = 1 is constant along a geodesic ray, then h = sin r along that ray. (Note that h cannot be zero as ω2 = h dr, so the exponential map is not a diffeomorphism and geodesic polar coordinates are not defined for r ≥ π. Similarly, if K = 0 along a geodesic ray, then h = r along that ray, just as on a flat plane. Similarly, if K = −1 along a geodesic ray, then h = sinh r along that ray. A length preserving diffeomorphism of surfaces is a diffeomorphism preserving the lengths of all curves. Theorem G.10. Any surface of constant Gauss curvature K = K0 is locally length preserving diffeomorphic to any other. Proof. In our geodesic normal coordinates, ω1 = dr, ω2 = h(r) dθ where k0 ..=

p

|K0 | and  sin k0 r   k0 , h(r) = r,   sinh k0 r k0

,

if K0 > 0, if K0 = 0, if K0 < 0.

Matching up the values of r, θ on our two surfaces then matches up ω1 , ω2 and so up lengths of any curves, as the length of a curve (r(t), θ(t)) is R pmatches ω12 + ω22 . More generally, the function h(r, θ) is constant in θ just when the surface admits “rotations”, i.e. rotation of θ, as symmetries, fixing the point about which the geodesic coordinates are computed. One easily checks that the surface is then length preserving diffeomorphic near that point to a surface of revolution, with that point fixed under the revolution: an intrinsic description of the shape of a surface of revolution. The length of the circle of radius r around the fixed R point is `(r) = ω2 = 2πh(r), and the curvature is K = −`00 (r)/`(r): two surfaces of revolution are length preserving diffeomorphic near fixed points just when their curvatures are equal as functions of distance from fixed point.

Geodesics are locally minimal Theorem G.11. Geodesics are locally minimal, and every locally minimal curve is a geodesic. Proof. In geodesic polar coordinates, r increases at unit rate as we move along the radial curves given by constant θ, and at less than unit rate as we move at unit speed in any other direction. Note that these are geodesics, by definition of the coordinates. Hence the length of a piece of curve increases by at least the increase in r across its end points, and exactly by this amount precisely for the radial curves. So the shortest curves of given increase in r, i.e. shortest

140

Geodesics on surfaces

paths between circles of constant r, are these radial curves. For any path, any two points on it sufficiently close together are each in the injectivity radius of the other, so the geodesic polar coordinates are defined, and the unique shortest path between those points is the unique geodesic between them in those coordinates. The length metric distance between two points p, q ∈ S of a surface S in E3 is the infimum of lengths of paths from p to q. As above, the geodesics between sufficiently close points have minimal length, so their length achieves the length metric distance. G.5 Prove that if the length metric distance between points goes to zero, then the distance in E3 also goes to zero. Corollary G.12. Every invertible map ϕ : P → Q of surfaces P, Q in E3 which preserves lengths of curves is a diffeomorphism. Proof. The map takes geodesics to geodesics, since they are precisely the locally shortest paths. The map identifies each point p0 ∈ P with a point q0 ∈ Q, and the circle around p0 of a given radius with the corresponding circle around q0 of the same radius. The lengths of geodesic segments are preserved, so if a point moves at unit rate along a geodesic on P then its image in Q does the same. Indeed, the same is true for a point moving along any curve (for example, along a circle), because the path is approximated by piecewise geodesic paths. Therefore at every point near p0 , except at p0 itself, ϕ∗ e1 = e1 and ϕ∗ e2 = e2 . Therefore ϕ is continuously differentiable at all those points. By lemma G.8 on page 137, every point p ∈ P is near enough but not equal to some such point p0 to use this argument to prove that ϕ is continuously differentiable at p. The same applies to ϕ−1 . Corollary G.13. On any connected surface S in E3 the following are equivalent: a. The surface S is complete in the length metric, i.e. length metric Cauchy sequences converge. b. The surface S is proper in the length metric, i.e. all length metric balls are compact. c. The exponential map on S has flow defined for all time through all points. d. Every geodesic on S defined on an open interval extends uniquely to a geodesic defined on a closed interval. e. There is a point x0 ∈ S so that every shortest path x : [a, b) → S with x(a) = x0 extends uniquely to a continuous path x : [a, b] → S. In particular, if S is a closed subset of E3 , then all of these hold.

141

Theorema Egregium

Proof. Every geodesic on S, parameterised by arc length, is defined on a maximal open interval, since the geodesics are the solutions of an ordinary differential equation. We recall some facts about metric spaces [14]. A length space is a metric space in which the distance between points is the infimum of lengths of paths between them. A minimal geodesic in a length space X is a path x(t) on which the distance between points x(t0 ), x(t1 ) is |t1 − t0 |. A geodesic is a path which is locally a minimal geodesic. By the Hopf–Rinow theorem for metric spaces [14, 27], any locally compact length space is complete just when any geodesic defined on an open interval extends to be defined on a closed interval. In any complete surface, geodesics then extend further to be defined on larger open intervals, since they continue to solve their differential equation. So if S is complete, geodesics are defined for all time. But if geodesics are defined for all time, they extend, so by Hopf–Rinow S is complete. G.6 Prove that, on any complete surface, vector fields of bounded norm are complete. Corollary G.14. Any two local isometries f, g : P → Q of the length metric between two connected surfaces are identical just when there is a point p0 ∈ P at which both f (p0 ) = g(p0 ) and f 0 (p0 ) = g 0 (p0 ). Proof. Each vector v ∈ Tp0 P is taken to the same vector f 0 (p0 )v = g 0 (p0 )v, so exponentiates to the same point of Q, if v is small enough. Because f and g are local isometries, they locally identify the geodesics of P and Q, and the exponential maps, so f ◦ expp0 = expq0 ◦f 0 (p0 ) = expq0 ◦g 0 (p0 ) = g ◦ expp0 , and since the exponential map is a local diffeomorphism, f = g near p0 . So the set of points where f = g is both open and closed.

Theorema Egregium A circle on a surface is the set of points at given length metric distance from a given point of the surface. We would like to see that the Gauss curvature controls the speed at which circles grow with radius. Along each geodesic coming out of a given point o, erect a frame with e1 tangent to the geodesic and e2 normal, a point of the frame bundle associated to each point near (but not equal to) o. Motion along e1 takes us away from o at unit speed, while motion along e2 spins us around the circles around o at unit speed. h(r, θ) = r −

K(o) 2 2 r + o(r) . 2

The length of the circular arc of radius r about o between two angles θ0 , θ1 is   Z Z θ1 K (o) r 2 ω2 = h(r, θ) dθ = |θ1 − θ0 |r 1 − + o(r) . 2 θ0

142

Geodesics on surfaces

Theorem G.15 (Theorema Egregium). Any isometry of length metric between analytic surfaces is analytic and preserves Gauss curvature. Proof. Distances are preserved, so circles (points of given distance) are preserved. Lengths are preserved, so lengths of circles are preserved, so Gauss curvature is preserved. Geodesics coming out of a given point are mapped to geodesics out of the corresponding point. The geodesics coming out of o have constant values of θ, and these are preserved. So θ changes under isometry only as a function of θ, independent of r. Since r is preserved, lengths of circular arcs are preserved, so the absolute differences of values of θ are preserved. So all isometries have the form r, θ 7→ r, ±θ + c, for some constant c. Arrange by choice of coordinates that the rays θ = 0 are matched, and the direction of increasing θ is matched: r, θ 7→ r, θ. G.7 Suppose that S is a surface in E3 diffeomorphic to a triangle in the plane, say with interior angles α, β, γ, and with geodesic sides (a “geodesic triangle”). Prove that Z α+β+γ =π+ K dA. S

In particular, negatively (positively) curved geodesic triangles have larger (smaller) angle sums than Euclidean triangles.

The Hessian We will prove: Theorem G.16. Take an open set on a surface in E3 , with smooth boundary of nonnegative geodesic curvature. Whenever a geodesic enters the interior of that open set, it is not tangent to the boundary. Take a function f : S → R. Pulling f back to the frame bundle of S, because df is semibasic, df = f1 ω1 + f2 ω2 for some functions f1 , f2 . Differentiate to find         f 0 α f1 f11 f12 ω1 d 1 + = f2 −α 0 f2 f21 f22 ω2 for unique functions fij = fji . Check that the operator D2 f ..= fij ωi ωj is a symmetric bilinear form on tangent spaces of S, the Hessian of f . The differential df restricts to any curve to become df =

df ds = f1 ω1 ds

measuring the rate at which f increases along the curve, and the Hessian becomes d2 f df1 = = f11 , 2 ds ds

143

Abstract surfaces

the acceleration of f along the curve. Any curve on which the Hessian is positive has f accelerating, so at any point of that curve, if df vanishes at that point of the curve, i.e. the curve is tangent to a level set of f , then f increases on either side of that point, i.e. the curve stays outside of the sublevel set. If df 6= 0, we can pick out a particular choice of frame: make e1 tangent to level sets of f . So df = f2 ω2 , i.e. f1 = 0, on this subbundle of the frame bundle, adapted to f . We can further orient the surface by requiring f2 > 0. Our equations simplify to f2 α = f11 ω1 + f12 ω2 , df2 = f21 ω1 + f22 ω2 . On any one level set ω2 = 0 so α = −κ2 ω1 = −

f11 , f2

so the geodesic curvature of the level set is κ2 =

f11 , f2

so the sign of the geodesic curvature of each level set is the sign of the acceleration of f along that level set. In particular, the level sets are geodesic just when the Hessian vanishes along all of them. Given any curve C, we can locally make a function f with that curve as level set. Take two curves C0 and C1 tangent at a point. Suppose they are both oriented, and lie on an oriented surface. Write each as a level set of functions f0 and f1 , with differentials pointing on the positive side of the oriented tangent line. At the point of tangency, both functions have vanishing differential on both curves. Clearly f0 has larger Hessian on that tangent line just when C0 has larger curvature. So then f0 − f1 has positive Hessian on that tangent line, and so f0 − f1 grows nearby on C1 . But f1 is constant on C1 . So f0 grows on C1 at all points nearby. So C1 has just one point of tangency with C0 near that point, and everywhere else nearby leaves the sublevel set of f0 . In particular, if C1 is a geodesic, f1 has vanishing Hessian along C1 , proving theorem G.16 on the facing page. G.8 Suppose that M is a complete surface in 3-dimensional Euclidean space, and f is a smooth proper function on M whose only critical point is a single local minimum. Suppose that all level sets of f are positively curved. Prove that every geodesic on M enters each compact set only finitely many times.

Abstract surfaces A surface is a manifold of dimension 2. A Riemannian metric on a surface is a choice of inner product in each tangent space, so that inner products of locally

144

Geodesics on surfaces

defined analytic vector fields are analytic functions. For simplicity of notation, write inner products as dot products. An orthonormal frame (x, e) is a choice of point x of S and a choice of orthonormal basis e1 , e2 of the tangent plane Tx S. The tangent frame bundle of S is the collection of orthonormal frames of tangent spaces of S. Note that for a surface S in E3 , this is not the same as the frame bundle we constructed previously in appendix D on page 101, which we could refer to as the bundle of ambient adapted frames. The tangent frame bundle of a surface in E3 is the quotient of the ambient adapted frame bundle by ignoring e3 , giving a 2 − 1 covering map. Nonetheless, we will use the same notation ⌜S for the tangent frame bundle. Take any two linearly independent vector fields on an open subset U ⊆ S of our surface. Apply the Gram–Schmidt orthogonalization procedure to them to construct orthonormal vector fields, say e01 , e02 , defined on U . Over U , any orthonormal frame e1 , e2 is uniquely expressed as e1 = cos θ e01 ± sin θ e02 , e2 = − sin θ e01 ± cos θ e02 . In this way, we identify ⌜U ∼ = U × (S 1 ∪ S 1 ), so the tangent frame bundle becomes a manifold. Let p : (x, e) ∈ ⌜S 7→ x ∈ S. The soldering forms ω1 , ω2 on ⌜S are defined by p0 (m, e1 , e2 )v = ω1 (v)e1 + ω2 (v)e2 . If the dual 1-forms to e01 , e02 on U are ω10 , ω20 then     0 ω1 cos θ ± sin θ ω1 = . ω2 − sin θ ± cos θ ω20 Suppose henceforth that the surface S is oriented. We pick out only the positively oriented frames, to get rid of ± so (using the same notation to also denote the bundle of positively oriented orthonormal frames) ⌜U ∼ = U × S 1 . Let . ω .= ω1 + iω2 , and write this as ω = eiθ ω 0 . The area form dA ..= ω10 ∧ ω20 pulls back to dA = ω1 ∧ ω2 , independent of the choice of e01 , e02 , so globally defined on S. Since dA is a nonzero area form, dω10 , dω20 are multiples of it, say dω10 = −a1 dA, dω20 = −a2 dA. Let α0 ..= a1 ω10 + a2 ω20 . Take exterior derivative of both sides to see that dω 0 = iα0 ∧ ω 0 on U , and that dω = i (α0 + dθ) ∧ ω.

145

Abstract surfaces

The real-valued 1-form α0 on U is uniquely determined by 0 = dω 0 − iα0 ∧ ω 0 . So the real-valued 1-form α ..= α0 + dθ is as well, giving 0 = dω − iα ∧ ω for a unique real-valued Levi-Civita connection 1-form α. Take exterior derivative of that equation to see that dα = Kω1 ∧ ω2 , for a unique function K on the tangent frame bundle. But on ⌜U , dα = d(α0 + dθ) = dα0 is the same for any choice of framing, i.e. K is a function well defined on the surface S. G.9 Suppose that S is not orientable. Prove that K is nonetheless well defined on S, while the area form dA = ω1 ∧ ω2 is not. Prove that the curvature form K dA is well defined on S if and only if K = 0. Theorem G.17 (Theorema egregium). The Levi-Civita connection 1-form and the Gauss curvature of a Riemannian metric on a surface depend only on the Riemannian metric, not on any choice of isometric immersion into Euclidean space. Geodesic normal coordinates work out exactly the same. But on abstract surfaces, we can reverse their construction. Take any function K(r, θ); there is a unique solution h to hrr + Kh = 0 with h = 0 and hr = 1 at r = 0: every function K defined near the origin of the plane is the Gauss curvature of some abstract Riemannian metric dr2 + h2 dθ2 near the origin. G.10 Prove that this metric is smooth at the origin. If K < 0 then hrr = −Kh > 0 for h > 0, so h > r; if K is defined in the entire plane, r, θ are geodesic normal coordinates, so this metric is complete. Theorem G.18. Any connected surface bearing a complete metric of negative Gauss curvature has exponential map at any point a smooth universal covering map from the plane. Proof. The exponential map is defined in the plane by completeness, but h has no zeroes so the exponential map is a local diffeomorphism. The pullback metric is complete: ω12 + ω22 = dr2 + h2 dθ2 ≥ dr2 + r2 dθ2 . Take a closed disk on the surface, say of radius r. Its preimage is a union of disks in the pullback metric, closed by completeness. Each sits in a Euclidean disk of smaller radius. If, no matter how small we make r, some of these disks overlap, then there are points of arbitrarily small Euclidean distance mapping to the same point of the surface. Take a convergent sequence of these, a contradiction. So the exponential map covers each small enough embedded closed disk on the surface with disjoint embedded closed disks, and so is a covering map. Smoothness of isometries and the Gauss–Bonnet theorem follow, with the same proof, for any oriented surface with Riemannian metric.

Au g u s t i n L o u i s C au c h y

S o f i a Va s i ly e v n a Kova l e v s k aya

É l i e C a rta n Erich Kähler

S h i i n g - S h e n C h e r nb

M a s at s u g u K u r a n i s h i Ro b e rt B rya n t a

P h i l l i p G r i f f i t h sc

Hints

Il ne me suffit pas de lire que les sables des plages sont doux ; je veux que mes pieds nus le sentent . . . Toute connaissance que n’a pas précédée une sensation m’est inutile. It is not enough for me to read that the sand on the seashore is soft. My bare feet must feel it . . . All knowledge that is not preceded by a sensation is useless to me. — André Gide Les nourritures terrestres

1.1. a. All submanifolds. b. Any discrete set of points. c. Take a 0-dimensional submanifold of the image. Its preimage is a submanifold. Take any submanifold of that submanifold. d. If dx1 ∧ dx2 6= 0, these are locally yi = the Lagrangian surfaces.

∂S ; ∂xi

in general the integral surfaces are

e. If dx 6= 0 these are y = y(x), z = dy/dx; in general they are Legendre curves. 1.2. a. All linear subspaces of all tangent spaces. b. The zero subspaces of all tangent spaces. c. The linear subspaces tangent to the fibers. d. If dx1 ∧ dx2 6= 0, these are dyi = aij dxj with aij = aji ; in general they are zero subspaces, lines, or Lagrangian planes. e. If dx 6= 0 these are dy = z dx, dz = a dx for any real numbera. 1.4. M = R, I = (x dx), X = { 0 } 1.8. Take any local coordinates on M , say xi , ua , near a point m0 . Every p-plane E ⊂ Tm M , with m near m0 , and with dx1 ∧ · · · ∧ dxp 6= 0 on E, is the set of tangent vectors on which dua = qia dxi , for uniquely determined numbers qia . As a short hand, write this as du = q dx. The functions xi , ua , qia on Grp M are local coordinates. The map (m, E) ∈ Grp M 7→ m ∈ M is (x, u, q) 7→ (x, u), a submersion. We leave the change of coordinates to the reader. 1.9. If θ = du − q 2 dx then dθ = −2q dq ∧ dx. 1.12. Suppose that, to each point x ∈ M of a manifold M , we have an associated p × q matrix ϕ(x), with entries analytic functions on M . Let K be the set of pairs (x, v) so

149

150

Hints

that x ∈ M and v is a vector in the kernel of ϕ(x). If ϕ(x) has rank independent of x, let us prove that K is an embedded submanifold of M × Rq . Let’s also prove that, for any map w : M → Rp , the following are equivalent: a. At every point x ∈ M , there is a vector v so that ϕ(v) = w(x). b. Near every point x ∈ M , there is an analytic vector valued function v on an open subset of M , so that ϕ(v(x)) = w(x); if ϕ is 1-1 at every point x ∈ M , then v is unique. We can change ϕ by multiplying by arbitrary invertible matrices of analytic functions. Since ϕ has constant rank, for any chosen point of M , we can permute rows and columns to get the upper left corner to be invertible at that point, and hence near that point, of the same rank as ϕ:

 ϕ=

A C

B D

 ,

with A invertible. Multiply by





A−1 0

0 , I

to get A = I. Having rank exactly that of the upper left corner is precisely D = CB. Multiply:       I 0 I −B I 0 ϕ = . −C I 0 I 0 I The reader familiar with vector bundles [6] may generalize. 1.13. We prove it in each tangent space separately. Real analyticity then follows from the uniqueness of solutions of the linear equations by problem 1.12 on page 6. So we prove it for constant coefficient 1-forms in Rp . Choose basis in Rp so that in coordinates x1 , x2 , . . . , xk , y1 , y2 , . . . , yp−k

P

P

X

bi` dy` ∧ dxi ,

our 1-forms are ξi = dxi . Write αi = aij dxj + 0 = α1 ∧ ξ1 + α2 ∧ ξ2 + · · · + αk ∧ ξk is precisely 0=

X

aij dxj ∧ dxi +

j

=

bi` dy` , say. The equation

`

X 1X (aij − aji ) dxj ∧ dxi + bi` dy` ∧ dxi , 2 j

`

Plugging in the unit vectors pointing along various coordinate axes, we find aij = aji and bi` = 0. 1.16. This is already clear geometrically: the shape operator is the curvature of geodesics. Let M ..= ⌜E3 be the frame bundle of E3 . The frame bundle ⌜S of any surface S in E3 satisfies ω3 = 0 and



γ13 γ23



 =

a11 a21

a12 a22

  ω1 ω2

for some smooth functions a11 , a12 = a21 , a22 . To have vanishing shape operator, 0 = a11 = a12 = a22 , so 0 = ω3 = γ13 = γ23 .

151

Hints

Consider on M the exterior differential system with equations ϑ0 = ω3 , ϑ1 = γ13 , ϑ2 = γ23 . The frame bundle ⌜S of any surface with vanishing shape operator is a 3-dimensional integral manifold. To be precise, each component of ⌜S is an integral manifold, since S might have more than one orientation, so ⌜S might have more than one component. Check that the exterior differential system is Frobenius, so there is a unique integral manifold through each point of M . But we already have an example of such an integral manifold, so that must be the only one. 1.17. Again take M = ⌜E3 with the exterior differential system ω3 , γ13 − c0 ω1 , γ23 − c0 ω2 , Again the system is Frobenius, so there is only one 3-dimensional integral manifold through each point. Rotate and translate a sphere of suitable radius to see one such integral manifold through each point of M , the frame bundle of that sphere. 1.18. Problem 1.17 on page 8 handles the case of one principal curvature, so assume that there are two principal curvatures. After picking a frame on a surface S so that e1 and e2 are in the principal directions, we find 0 = ω3 = γ13 − k1 ω1 = γ23 − k2 ω2 , for constants k1 , k2 . Differentiate all three equations to find that 0 = (k1 − k2 ) γ12 ∧ ω2 = (k1 − k2 ) γ12 ∧ ω1 , forcing γ12 = 0 by Cartan’s lemma. Differentiating the equation γ12 = 0, we find 0 = k1 k2 ω1 ∧ ω2 so that either k1 = 0 or k2 = 0. We can assume that k1 = 0, i.e. γ13 = 0. But then de1 = (e1 · de1 ) e1 + (e2 · de1 ) e2 + (e3 · de1 ) e3 , = γ11 e1 + γ21 e2 + γ31 e3 , = 0. Therefore e1 is constant as we travel along the surface. So the surface is a collection of straight lines, in this e1 direction, all placed perpendicular to a curve in the e2 , e3 plane. On that curve, ω1 = 0, and dω2 = 0 so we can write locally ω2 = ds for some function s. Then we find de2 = (e1 · de2 ) e1 + (e2 · de2 ) e2 + (e3 · de2 ) e3 , = γ12 e1 + γ22 e2 + γ32 e3 , = −k2 dse3 . and de3 = (e1 · de3 ) e1 + (e2 · de3 ) e2 + (e3 · de3 ) e3 , = γ13 e1 + γ23 e2 + γ33 e3 , = k2 dse2 . Check that the vectors E2 = k2 cos(s)e2 + k2 sin(s)e3 , E3 = −k2 sin(s)e2 + k2 cos(s)e3 are constant. Rotate so that E1 = e1 , E2 , E3 are the standard basis vectors of E3 , to see that the surface S is a right circular cylinder of radius 1/ |k2 |. 1.22. Start by looking at the geometry of the given foliation. Suppose that the foliation is of an open subset U of 3-dimensional Euclidean space. Consider the 4dimensional manifold M of all orthonormal frames e1 , e2 , e3 at points x ∈ U for which e3 is perpendicular to the leaf through x of the given foliation. By the Frobenius

152

Hints

theorem, 0 = ω3 ∧ dω3 , = ω3 ∧ (γ13 ∧ ω1 + γ23 ∧ ω2 ), = γ13 ∧ ω1 ∧ ω3 + γ23 ∧ ω2 ∧ ω3 . So on M ,



γ13 γ23





a11 a21

=

a12 a22

a13 a23

 ω1 ! ω2 ω3

for some functions aij on M . Plugging these in above, we find that a12 = a21 . Clearly a11 ω12 + a12 ω1 ω2 + a21 ω2 ω1 + a22 ω22 is the shape operator of each leaf of the foliation. Now consider the problem of constructing a triply orthogonal web incorporating this foliation as one of its three. We need to find a choice of e1 , e2 at each point to construct a frame, so that e1 will be perpendicular to the leaves of the first foliation, and e2 perpendicular to the leaves of the second foliation, and the third foliation will be the one we started with, already perpendicular to e3 . So we will need to solve the exterior differential system 0 = ω1 ∧ dω1 = ω2 ∧ dω2 on M , i.e. with the equations



γ13 γ23



 =

a11 a21

a12 a22

a13 a23

 ω1 ! ω2 ω3

already in force. Note that on M , ω1 , ω2 , ω3 , γ12 are linearly independent 1-forms. We are looking for an integral 3-manifold X of that exterior differential system, on which we want ω1 , ω2 , ω3 to be linearly independent, i.e. X projects by local diffeomorphism to 3-dimensional Euclidean space. It might be simpler to write our foliation shape operator as



γ13 γ23



 =

a131 a231

a132 a232

a133 a233

 ω1 ! ω2 ω3

and then we are imposing only the relations a132 = a231 , i.e. symmetry in these two outer indices. Differentiate the equations of our exterior differential system to find that on any integral 3-manifold X: 0 = ωi ∧ dωi , = −ωi ∧

X

γij ∧ ωj ,

j

which forces γij to be a linear combination γij =

X k

aijk ωk ,

153

Hints

with aijk = −ajik since γij = −γji . But plug in to get 0 = ωi ∧

X

(aijk − aikj )ωk ∧ ωj

jk

so that aijk = aikj if i, j, k are all distinct. So for distinct indices, aijk is symmetric in jk, but antisymmetric in ij. These two involutions generate the permutation group, and so the sign of aijk is a representation of the permutations on 3 letters. But any two involutions are conjugate in the permutation group, so they must force the same sign change. Hence aijk = 0 for i, j, k distinct. This is precisely the demand that the shape operators of the leaves of all three foliations are thus diagonal in the frame e1 , e2 , e3 . In other words, each leaf of each foliation lies normal to each leaf of each other foliation, and intersects tangent to a principal direction, i.e. along a principal curve. We can assume that U is connected. Either a. the leaves of the given foliation are everywhere umbilic, hence each leaf is an open subset of a sphere or plane, and so a132 = 0 and a131 = a232 , or else b. the vectors e1 , e2 have to be chosen in principal directions and this determines them up to 4 choices, at least on a dense open subset of U . Suppose that the leaves are spheres or planes, so a132 = 0 and a131 = a232 , say. The exterior differential system is generated in dimension 3 by γ12 ∧ ω1 ∧ ω2 = 0, so every line or plane in any tangent space of M is an integral element. Planes, i.e. integral planes, generically have ω1 , ω2 linearly independent on them, so are generically of the form γ12 = p1 ω1 + p2 ω2 , ω3 = q1 ω1 + q2 ω2 ,

so lie in a unique 3-dimensional integral element γ12 = p1 ω1 + p2 ω2 . The polar equations of any integral point or line are trivial, but the generic integral plane has polar equation γ12 − p1 ω1 − p2 ω2 , so s1 = 0, s2 = 1, s3 = 0, solutions depend on 1 function of 2 variables. We can explicitly construct the web: take any leaf of our given foliation, the initial leaf , and draw on it any foliation by curves. On that same surface, draw the orthogonal foliation by curves. Drag each leaf of each of those foliations along the flow of e3 , through space, to trace out a surface. We want to see that this construction always creates a triply orthogonal web. Any triply orthogonal web containing the given foliation has to arise from this construction: the leaves of the other two foliations intersect the initial leaf in curves, and the vector field e3 is tangent to every leaf of the other two foliations. Conversely, take a foliation of the initial leaf by curves. Locally pick any orthonormal vector fields e1 , e2 tangent to that leaf, with e1 tangent and e2 perpendicular to that curve foliation. Drag, as above. We produce two more foliations of Euclidean space, defined near that leaf. But it is not clear whether they remain perpendicular. The leaves of the two new foliations both contain e3 , and they start off perpendicular along the initial leaf. Compute the change in the Euclidean metric along the flow of e3 : Le3 ωi ωi = 2ai3j ωi ωj .

154

Hints

So umbilicity of the given leaves is precisely the condition that the Euclidean metric varies only by scaling as we flow along e3 , on the perpendicular vectors to e3 , and so any pair of planes in a tangent space which start perpendicular will remain so. Hence the construction always succeeds. Suppose now that the leaves are nowhere umbilic. We will see that each foliation by nonumbilic surfaces lies in at most one, and typically no, triply orthogonal web. We need e1 , e2 to diagonalize the shape operator, i.e. to lie in principal directions. Imagine drawing the principal curves, i.e. the curves in those directions, on each leaf. Our given foliation by surfaces has now, on each surface, two foliations by perpendicular curves. We want to see whether, when we flow along the unit normal vector field e3 of the foliation, these principal curves flow into one another. For a generic foliation by surfaces, the flow of e3 will spin the principal curves of one leaf around into various curves on other leaves, not necessarily principal. We need a132 = 0, so the shape operator is diagonalized, i.e. e1 , e2 point in principal directions, and we suppose the leaves are not umbilic, so a131 6= a232 . Our frames form a 3-manifold X in the P frame bundle. We have to decide whether X is an integral manifold. On X, γij = k aijk ωk , for some functions aijk = −ajik . To have a triply orthogonal web, we need precisely that aijk = 0 if i, j, k are distinct, i.e. the shape operators are diagonalized. This is precisely the condition that the prinicipal curves flow into one another. 2.1. We can ask that e1 be perpendicular to the leaves of our foliation. (If you use e3 instead of e1 here, you will find it more complicated to write out the tableau.) So then ω1 ∧ dω1 = 0 on the foliation. Expand out dω1 = −γ12 ∧ ω2 − γ13 ∧ ω3 to arrive at the tableau



γ12 γ13



ω1 ∧ ω2 ω1 ∧ ω3

! .

so s1 = 0, s2 = 1, s3 = 1, foliations of open sets of 3-dimensional Euclidean space depend on 1 function of 3 variables. We can see them as the level sets of 1 function of 3 variables, but the function is defined only up to composition with a strictly increasing or strictly decreasing function. 2.2. The polar equations of Ej are given by setting ω i = 0 for i > j, and plugging in, with π α = 0 on Ej , so all terms with 2 or more π vanish, i.e. the polars in grades 0, 1, 2, . . . , j. Hence the characters of Ep are the numbers of polars in each grade. 2.5. Take R15 with coordinates xi , ui , uij for i, j = 1, 2, 3. Note that our differential equations, spelled out as algebraic equations u32 − u23 = f 1 − u1 , u13 − u31 = f 2 − u2 , u21 − u12 = f 3 − u3 , cut out a submanifold M ⊂ R15 of dimension 12. Take ω i ..= dxi , θi ..= dui − uij dxj , i

∂f i and πji ..= duij . It will help to denote ∂x j by fj . The equations of M force 3 linear i relations among the πj :  du32 − du23 = fi1 − u1i dxi ,

155

Hints modulo θ1 , θ2 , θ3 , and so on, i.e. π32 = π23 − fi1 − u1i ω i ,



π31 = π13 + fi2 − u2i ω i ,



π21 = π12 − fi3 − u3i ω i .



Our tableau: modulo θ1 , θ2 , θ3 ,

 1

π11

π21

θ  2 d θ 2  = −   π1 3 θ π13

π31

 1 ω  2  π32  ∧ ω  3

π22 π23

π11

π12

 2 = −  π1

π22

π13

ω

π33 π13

 1  1 ω τ  3  ∧ ω 2  + τ 2  π2  3

π23

0

ω

π33

where the torsion is

 1 τ τ2

 =

f13 − u31 0



ω 12 +



u21 − f12 f11 − u11





u22 − f22 + u33 − f33 f21 − u12

ω 13 +



ω 23

We can try to absorb torsion, for example by using

 

π101 π11 0 02 . π1  .= π12  + u31 − f13 π202 π22 0



u21 − f12 ω1 0  ω 2  , 1 f2 − u12 ω3

0 0 0

which we denote as π instead of π 0 to simplify notation. Our tableau: modulo θ1 , θ2 , θ3 ,

π11 θ  2 d θ 2  = −   π1 θ3 π13

 1

π12

π13

  1  ω uii − fii  23 2  π23  ∧ ω  +  0  ω . 3

π22 π23

0

ω

π33

The torsion is uii − fii (Einstein notation: implicitly summed over i). Take the submanifold M 0 ⊂ M cut out by the equation uii = fii . For simplicity, denote this submanifold as M henceforth. On M , i 0 = d(uii − fii ) = πii − fij ωj .

Our tableau: modulo θ1 , θ2 , θ3 ,

π11 θ  2 d θ 2  = −   π1 3 θ π13

 1

Let



π103 π203

π12

π13

π22

π23

π23

i −π11 − π22 + fij ωj



 3 =

π1 π23

 −

i fi1 i fi2



 1 ω   ∧ ω 2  .  3

ω3 ,

ω

156

Hints and once again just write these as π instead of π 0 . Our tableau: modulo θ1 , θ2 , θ3 ,

 1

θ d θ2  = −  θ3

π11

π12

π13

π12

π22

π23

π13

π23

−π11 − π22

s1

s2

s3

3

2

0

ω1  ∧ ω 2  .  ω3

Integral elements πji = pijk ω k . We highlight certain coefficients, to be discussed in chapter 5: p112 = p211

(1)

p113 = p311

(2)

p213 = p312

(3)

p212

= p221

(4)

p213

p321

(5)

p312 = p321

(6)

p313 = −p111 − p221

(7)

p323 = − p112 − p222 .

(8)

=

These coefficients are solved for in terms of others, except for the 3rd and 8th equations. But we use the 6th equation to fix up the 3rd, and the 1st equation to fix up the 8th, to solve for highlighted coefficients in terms of others. Hence the space of integral elements at each point of the 11-dimensional manifold M 0 has dimension given by counting the other coefficients: 7 dimensions of integral element at each point. Involution, with the general solution depending on 2 functions of 2 variables. i 2.8. qjk = cim` (gkm gj` − gjm gk` )

2.11. Choose the third fundamental form so that the Gauss curvature and the squared mean curvature have linearly independent differentials. Replace the surface by an open subset on which the Gauss curvature and the squared mean curvature are global coordinates invariant under rigid motion of the surface. Pick the eigenvalues so that the mean curvature is not zero; the surface is not symmetric under reflection in the tangent plane. So rigid motions fix every point of the surface, and also fix a normal direction, so are trivial. 3.5. If uµ , vµ are the vector fields dual to the real and imaginary parts of the ω µ , the torsion tensor, or Nijenhuis tensor T ..= (uµ + ivµ )τ ω µ 0,2 depends only on the almost complex structure, a section of ΩM ⊗ T M ⊗ C.

3.7. Consider left action of SU3 on itself: Lh z = hz. The identity function g(z) = z behaves like (L∗h g)(z) = g(Lh z) = g(hz) = hz = hg(z), so L∗h g = hg. Thus for any

157

Hints

constant matrix h ∈ SU3 , L∗h ω = L∗h (g −1 dg), = (L∗h g)−1 dL∗h g, = (hg)−1 d(hg), = g −1 h−1 h dg, = g −1 dg, = ω.

3.8. Differentiating ω = g −1 dg, i.e. dg = gω, 0 = dg ∧ ω + g dω, = gω ∧ ω + g dω, dω = −ω ∧ ω.

3.11. [36] pp. 36–37 4.1. Take an integral manifold X coframed by ω1 , ω2 , α. We need to prove that every point of X lies in an open subset of X which is also an open subset of a frame bundle of an isometric immersion. Note that X ⊂ M ..= ⌜S × ⌜E3 . We use a slightly imprecise notation: the map taking x, e1 , e2 , x0 , e01 , e02 , e03 7→ x



we will simply denote by x, and so on. At each point x, e1 , e2 , x0 , e01 , e02 , e03 ∈ X,



the vectors e1 , e2 ∈ Tx S are an orthonormal basis of Tx S. Since ω1 , ω2 are linearly independent on X, there are vectors tangent to X on which ω1 , ω2 take any given values. But ω1 = e1 · dx and ω2 = e2 · dx measure dot products of dx with e1 , e2 . So there are tangent vectors to X for which the map x : X → S has differential dx taking these vectors to have any given values of dot products with e1 , e2 . Hence x : X → S has rank 2, i.e. full rank, a submersion. We can replace X by an open subset of X, and S by an open subset of S, so that this submersion is a fiber bundle map with connected fibers. Moreover, ω1 , ω2 vanish just on the directions where dx = 0, i.e. the fibers. On X, ω10 = ω1 , ω20 = ω2 and ω30 = 0. So dx0 = 0 just on the tangent vectors to X on which 0 = ω10 = ω20 , i.e. on which 0 = ω1 = ω2 , i.e. on which dx = 0, i.e. on the fibers of x. So x0 is constant on each fiber of x. So x0 is defined and smooth on the base S of the fiber bundle map x : X → S, i.e. x0 factors through x. Write this factorization as x0 = ϕ(x), ϕ : S → E3 , a commutative diagram of maps: X x0

x

S

ϕ

E3 .

158

Hints

Compute x0∗ = ϕ∗ x∗ , = ϕ∗

X

(ei · x∗ )ei ,

i

=

X

ϕ∗ ei (ei · dx),

i

=

X

ϕ∗ ei ωi .

i

But x0∗ = dx0 , =

X

=

X

=

X

(e0i · dx0 )e0i ,

i

e0i ωi0 ,

i

e0i ωi .

i

So finally on X,

X i

e0i ωi =

X

ϕ∗ ei ωi .

i

By linear independence of ω1 , ω2 : e0i = ϕ∗ ei . Each point x ∈ S lies in the image of some point x, e1 , e2 , x0 , e01 , e02 , e03 ∈ X



and ϕ0 (x) : e1 , e2 7→ e01 , e02 , so ϕ0 (x) takes some orthonormal basis e1 , e2 to an orthonormal basis e01 , e02 . Hence ϕ is an isometric immersion. Moreover, X consists of isometrically matched orthonormal bases, i.e. X ⊂ Xϕ , an open subset since both X and Xϕ are 3-dimensional submanifolds of M . 5.3. Take a ϑ-integral element E0 = he1 , e2 , . . . , ep i . Move it: Et = e1 + tw1 + O(t)2 , . . . , ep + twp + O(t)2 .

Every motion through p-dimensional subspaces has this form locally. Expand: ϑ|Et = t

X i

ϑ(e1 , . . . , ei−1 , wi , ei+1 , . . . , ep ) + O(t)2 .

159

Hints

The differentials, i.e. linear terms, are sums of polar equations. Set all but one wi to zero: every polar equation is a differential. In coordinates, E0 = (du = 0), Et = (du = q(t) dx), let wi (t) ..= qia (t)∂ua . 5.4. If an integral is not ordinary, we can find integral elements nearby with characters “borrowed” downward, so larger s0 , or the same s0 but larger s1 , or some such. As p + s0 + · · · + sp = dim M , borrowing raises one character and lowers some later character, decreasing the dimension dim M + s1 + 2s2 + · · · + psp , of the submanifold containing all nearby integral elements. 5.5. Generic linear subspaces are regular, so ordinary, so involutive. The polar equations along a generic integral element are the same. 5.6. Clearly E⊥ is an ideal of differential forms, and homogeneous (i.e. the direct sum of its intersections with forms each degree). But E⊥ might not be d-closed. Take its d-closure E∨ ; it has the same integral manifolds as E⊥ . Suppose that there is an exterior differential system I with E among its integral elements: I ⊂ E⊥ ⊂ E∨ . Every I-integral manifold has tangent spaces from E, so an E⊥ -integral manifold, so an E∨ -integral manifold. So I, E⊥ , E∨ share the same integral manifolds. Suppose now that E is an open subset in the set of involutive maximal dimensional I-integral elements. Every element of E arises as the tangent space of an I-integral manifold, by the Cartan–Kähler theorem, so as the tangent space of a E∨ -integral manifold. So E lies among the E∨ -integral elements, which lie among the I-integral elements. The differential of any form vanishing on E lies in E∨ , so also vanishes on E. So E⊥ = E∨ . Consider an E∨ -integral element E. Being an I-integral element, if E lies close enough to a linear subspace E 0 of some element of E, then E lies in a maximal dimensional involutive I-integral element, so an element of E, so in a maximal dimensional E∨ -integral element. So E∨ is involutive near E. But E∨ -integral elements near elements of E are I-integral elements, so elements of E. 5.7. Pick a tableau at m0 which is generic in the usual sense: having maximal s1 among all tableaux for the same system, and maximal s2 subject to that s1 , and so on. Let I 0 ⊆ I be the exterior differential system generated by the θa and the rows of the tableau. It is possible that various forms in I vanish at m0 , and near m0 lie outside I 0 . Note that the characters of I 0 at any I-integral element are no larger than those of I since I 0 ⊆ I. Suppose now that I is involutive. The characters of I are locally constant by problem 5.5 on page 41. The characters of I 0 are at least as large as those of the tableau, which match those of I at m0 . So I 0 and I have the same characters, and both are involutive on all of their integral elements, with the same dimensions of spaces of integral elements. Hence they have the same integral elements where the tableau maintains those same characters. 5.8. “Generic” is just a condition on linear independence of π 1-forms, so holds locally. Torsion is absorbable at a point just when there is a torsion free integral element coframed by ω 1 , . . . , ω p . Absorbing torsion is solving a system of linear equations of constant rank; apply problem 1.12 on page 6

160

Hints 6.2. Take local coordinates x1 , x2 , . . . , xp , y 1 , y 2 , . . . , y q , where E is the graph of dy = 0. If ϑ = cIA dxI ∧ dy A , we will see that Lv ϑ|E =

∂v a ∂cI cIa dxIj + v a a dxI . ∂xj ∂y

Let (−1)I mean (−1)m if I consists of m indices. Write v = vj

∂ ∂ + vb b . ∂xj ∂y

Note that v cI dxI = (−1)J v i cJiK dxJK . Commuting with exterior derivative, ∂v i j dx + ∂xj ∂v a j Lv dy a = dx + ∂xj Lv dxi =

∂v i b dy , ∂y b ∂v a b dy . ∂y b

By the Leibnitz rule, Lv ϑ = v i

∂cIA I ∂cIA I dx ∧ dy A + v a dx ∧ dy A ∂xi ∂y a J



+ cJiKA dx ∧ I

∂v i j ∂v i b dx + dy ∂xj ∂y b B



+ cIBaC dx ∧ dy ∧



∧ dxK ∧ dy A

∂v a j ∂v a b dx + dy ∂xj ∂y b

On E, dy = 0 so Lv ϑ|E = v i

∂cI I ∂cI dx + v a a dxI ∂xi ∂y ∂v i j dx ∧ dxK ∂xj ∂v a j dx . + cIa dxI ∧ ∂xj + cJiK dxJ ∧

This is not quite the same as v dϑ|E = v i

∂cI I ∂cI dx + v a a dxI ∂xi ∂y ∂cI`k jIK + (−1)jI v ` dx . ∂xj

Write the tangent part of v as ∂ . ∂xi Let A be the linear map A : E → E given by v0 = vi

Aij =

∂v i ∂xj



∧ dy C .

161

Hints

and apply this by derivation to forms on E as (Aξ)(v1 , . . . , vk ) = ξ(Av1 , v2 , . . . , vk ) − ξ(v1 , Av2 , v3 , . . . , nvk ) + . . . . Then Lv ϑ|E = v 0

dϑ E + v a

∂cI I ∂v a j dx + Aϑ|E + cIa dxI ∧ dx . a ∂y ∂xj

Since 0 = ϑ|E = dϑ|E , we find Lv ϑ|E = v a

∂cI I ∂v a j dx + cIa dxI ∧ dx . a ∂y ∂xj

6.3. Take local coordinates x1 , x2 , . . . , xp , y 1 , y 2 , . . . , y q , where E is the graph of dy = 0. Write ϑ = cIA dxI ∧ dy A . Write X as the graph y = y(x), so ϑ|X =

X

cIA (x, y)dxI ∧

∂y a` j` ∂y a1 j1 dx ∧ · · · ∧ dx . j 1 ∂x ∂xj`

so

∂cI I ∂v a j I dx + c dx ∧ dx . Ia ∂y a ∂xj depends only on knowledge of the point m where we compute coefficients of ϑ and of E = Tm X, so that we drop dy terms. Lv ϑ|E = v a

6.4. In coordinates, the linearization of an exterior differential system I about an integral element E ⊂ Tm M does not involve any dy ∧

dy terms but depends on the terms with no dy and with one dy explicitly. But P = E ⊥ = hdy a i. 6.6. Lv ϑ|E = cIa dxI ∧

∂v a j dx ∂xj

6.8. Suppose E ⊂ E+ is a noncharacteristic hyperplane. By uniqueness of extension E+ , the polar equations of E cut out precisely E+ inside Tm M , i.e. the dimension of polar equations of E is the dimension of Tm M/E+ . The polar equations of any regular hyperplane in E+ are satisfied on E+ , so have same rank. So the regular integral elements are also noncharacteristic. If p ..= dim E+ , the rank of polar equations of E is s0 + s1 + · · · + sp−1 , while the dimension of Tm M/E+ is s0 + · · · + sp . So sp = 0 on E+ just when every regular hyperplane in E+ is noncharacteristic. 6.9. Any hyperplane E− ⊂ E is I-characteristic just when it lies in some other Iintegral element E 0 of same dimension as E. But then E 0 is also a J -integral element, I J so E− is J -characteristic: ΞE ⊂ ΞE . 6.10. At each point, there is a unique maximal integral element, so every hyperplane in it lies in that unique maximal integral element: ΞE is empty. 6.12. The choices of p-dimensional integral element arise from the semipositive grade coefficients: k coefficients of each polar in grade k, so ksk in that grade in all. So if sp = 0 then there is a unique p-dimensional integral element containing E. If E+ is involutive, then the semipositive grade coefficients pα i are arbitrary. In an adapted tableau, E is determined by pα for i < p, which are all of the coefficients just when i sp = 0. 6.13. In terms of the tableau we gave previously in solving problem 7.1 on page 54, the complex points in the projective plane satisfying the equations of the minors are √ √ [0, 1, 0], [i, 1, 0], [−i, 1, 0], [i, 0, 1], [−i, 0, 1], [1, 1, 2i], [1, 1, − 2i].

162

Hints

The real ones for the characteristic variety as defined above, i.e. just [0, 1, 0] corresponding to the hyperplane 0 = dx2 . 6.14. For the harmonic function exterior differential system I, 2 I(x,y,u = hΘ, dϑ, ϑ ∧ (dx, dy, dux , duy )i x ,uy )

has dimension 6. The characters are s0 , s1 , s2 = 1, 2, 0, summing to 3, as we see from the tableau ! ! dx duy −dux ∧ dy dux duy 2 So the symbol matrix is not square: s0 + s1 + s2 = 3 < 6 = dim I(x,y,u . x ,uy ) 0 Here is a way to “fix” this problem. Let I be the ideal generated by the 2-forms ϑ ∧ dx, Θ, dϑ. The tableau of I 0 is

ϑ

0

 dx

 −dux  ∧

  duy dux

!

dy

duy

with characters s00 , s01 , s02 = 0, 3, 0. But now 2

I 0 (x,y,ux ,uy ) = hΘ, dϑ, ϑ ∧ dxi has dimension 3. Take an I 0 -integral surface S coframed by dx, dy and containing an I-integral curve C on which dx 6= 0, say with y = y(x). On S, ϑ ∧ dx = 0 so ϑ = f (x, y) dx locally. On C, 0 = ϑ = f (x, y(x)) dx. So f (x, y(x)) = 0. On S, 0 = dϑ = df ∧ dx, so f = f (x) and f (x, y) = f (x) = f (x, y(x)) = 0, hence ϑ = 0 on S. We conclude that any I-integral curve C on which dx 6= 0 lies in a locally unique integral surface S. The problem with directly applying the Cauchy–Kovalevskaya theorem is that I-integral curves are not arbitrary curves. 7.1. The exterior differential system lies on an 11-dimensional manifold M with tableau modulo θ1 , θ2 , θ3

π11 θ  2 d θ2  = −   π1 3 θ π13

 1

π12

π13

π22

π23

π23

−π11 − π22

 1 ω   ∧ ω 2  .  3 ω

Denote our manifold as M 11 to indicate that it is 11-dimensional. Take the flag M08 ⊂ M19 ⊂ M210 ⊂ M 11 given by M08 ..= (0 = x1 = x2 = x3 ), M19 ..= (0 = x2 = x3 ), M210 ..= (0 = x3 ). We have ω i ..= dxi , θi = dui − uij dxj , and πji = duij modulo θa , ω i . Pick a submanifold R28 ⊂ M210 of codimension s2 = 2 given by equations u22 = u22 (x1 , x2 ), u32 = u32 (x1 , x2 ),

163

Hints a submanifold R14 ⊂ R28 ∩ M19 of codimension s1 = 3 given by equations u11 = u11 (x1 ), u21 = u21 (x1 ), u31 = u31 (x1 ), and a point R00 ⊂ R14 ∩ M08 of codimension s0 = 3 given by equations u1 = c 1 , u2 = c 2 , u3 = c 3 , for functions uij and constants c1 , c2 , c3 . Since there are no free derivatives to restrain in the final column of the tableau, there is no further restraining manifold. So R00 is a point. On R14 , the equations 0 = θ1 = θ2 = θ3 become ordinary differential equations for functions u1 , u2 , u3 of x1 , which have a unique solution through the point R00 . Check that on R14 , all of the tableau vanishes, so we don’t have to solve any other equations than 0 = θ1 = θ2 = θ3 to produce an integral curve. One R28 , the tableau expands out to give derivatives in x2 : ∂u11 ∂u21 = + u31 − f13 , ∂x2 ∂x1 ∂u21 ∂u32 = , ∂x2 ∂x1 3 3 ∂u1 ∂u2 = . ∂x2 ∂x1 The last two have right hand sides expressed in terms of the restraining manifold data: u21 (x1 , x2 ) = u21 (x1 ) + u31 (x1 , x2 )

=

u31 (x1 )

+

Z

∂u32 2 dx , ∂x1

Z

∂u32 2 dx . ∂x1

With these solved for, the first equation then solves: u11 (x1 , x2 ) =

Z 

∂u21 + u31 − f13 ∂x1



dx2 .

It is not clear that the graph of these functions is an integral surface. Finally, we expand out the tableau to find equations for derivatives in x3 : ∂u11 ∂x3 ∂u21 ∂x3 ∂u22 ∂x3 ∂u21 ∂x3 ∂u32 ∂x3

∂u31 , ∂x1 ∂u32 = , ∂x1 ∂u32 = + u12 − f21 , ∂x2 ∂u1 ∂u22 = − 11 − − ∂x ∂x1 ∂u1 ∂u22 = − 21 − + ∂x ∂x2 =

∂fii , ∂x1 ∂fii . ∂x2

164

Hints

Any 3-dimensional integral manifold will arise by solving this determined system, with initial data from the integral surface. The proof of the Cartan–Kähler theorem shows that this procedure always constructs a 3-dimensional integral manifold, which is not obvious. 7.2. Two proofs: a. Writing out I in local coordinates as differential equations:



∂ui ∂uj = some function x, u, ∂x>i ∂x≤j

 ,

as on page 42, I 0 yields only the subset of those differential equations which have the form   ∂ui ∂uj = some function x, u, ≤j . ∂xp ∂x Apply the Cauchy–Kovalevskaya theorem to these equations. b. The I 0 -polar equations on Ep−1 are precisely those of I. The I 0 -integral elements containing Ep−1 are therefore those of I. So Ep−1 lies in a unique integral element of I 0 , which is Ep . The characters of I 0 are s00 0

s01 0

... ...

s0p−2 0

s0p−1 s0 + · · · + sp−1

s0p 0

Apply theorem 6.2 on page 49. 7.4. For k ≤ p, if I k = 0 then I k−1 ∧ Ω 1 ⊆ I k , so 0 = I k−1 ∧ Ω 1 and so 0 = I k−1 . 7.5. If ω is identically zero, then I p−1 ∧ Ω 1 = 0 so I p−1 = 0. Assume that ω is not identically zero. Write X locally as X = (0 = f ) for some function f with df 6= 0. If ω = 0 at every point of X, replace ω by ω/f , and repeat until ω 6= 0 at some point of X. Replace X+ by the open subset of X+ containing a point of X at which ω 6= 0. 7.6. Since ei form a basis, [ei , ej ] = ckij ek for some functions ckij . Write e, eˆı , eˆıˆ to denote e1 , . . . , ep , e1 , . . . , eˆi , . . . , ep , e1 , . . . , eˆi , . . . , eˆj , . . . , ep . By lemma 7.5 on page 55, dφ(e) = (−1)i+1 ei (φ(eˆı )) +

X

φ([ei , ej ], eˆıˆ).

i e + e> e˙ = 0, i.e. e> e˙ = −e˙> e = −(e> e) ˙ > , i.e. e> e˙ is antisymmetric, with entries ei · e˙ j . Since e˙ 1 = ke2 , 0 k 0

>

e e˙ =

? ? ?

? ? ?

! .

By antisymmetry, >

e e˙ = for some t.

0 k 0

−k 0 t

0 −t 0

!

171

Hints

D.5. For any constants α, β, γ, let x1 x2 x3

α cos βs α sin βs γs

! ..=

! .

so x˙ 1 x˙ 2 x˙ 3

−αβ sin βs αβ cos βs γ

! =

! ,

|x| ˙ 2 = α2 β 2 + γ 2 . So we need this to equal 1. Clearly α = 0 or β = 0 is a line, and γ = 0 is a circle. We ignore those cases and so the equation α2 β 2 + γ 2 = 1 can be solved for any of these three constants in terms of the other two. We can reflect in the x1 x2 plane to arrange that α, β > 0 and in the x3 axis to arrange that γ > 0. We get −αβ sin βs αβ cos βs γ

e1 = so

!

−αβ 2 cos βs e˙ 1 = ke2 =  −αβ 2 sin βs  , 0 so k = αβ 2 and cos βs sin βs 0

e2 = −

! .

Differentiate β sin βs −β cos βs 0

e˙ 2 = so

!

βγ 2 sin βs  te3 = e˙ 2 + ke1 = te3 = −βγ 2 cos βs . αβ 2 γ Finally, t = ±βγ, and e3 = ±

γ sin βs −γ cos βs αβ

So if we set β ..=

p

k 2 + t2 ,

k , β2 t γ ..= ± , β α ..=

! .

172

Hints

we get a helix with prescribed constant values of k, t. D.6. If e02 = e2 , then e1 , e3 and e01 , e03 are orthonormal bases of the same plane e⊥ 2 . Choose the sign of e3 to get e01 + ie03 = eiθ (e1 + ie3 ) for some angle θ. Differentiate to get ˙ iθ (e1 + ie3 ) + eiθ (e˙ 1 + ie˙ 3 ). de01 + i de03 = θe Expand to get ˙ iθ (e1 + ie3 ) + eiθ (k − it)e2 (k0 − it0 )e2 = θe Differentiate e2 = e02 to get −k0 e01 + t0 e03 = −ke1 + te3 . Hence k0 + it0 = e−iθ (k + it). So θ˙ = 0, a constant rotation. On the other hand, we can now pick any constant θ, and define: e01 + ie03 ..= eiθ (e1 + ie3 ), e02 ..= e2 , k0 + it0 ..= e−iθ (k + it), 0

x =

Z

e01 ds

and check that the Serret–Frenet equations are satisfied, so these yield the Serret– Frenet frame of a curve. D.8. The proof requires some unwinding of notation: the expression ωi = ei · dx P means that ωi = j eji dxj , which allows us to unwind the following formal steps: dωi = d (ei · dx) , = dei ∧ dx, =

X

(ej · dei ) ∧ (ej · dx) ,

j

=

X

γji ∧ ωj .

j

Similarly, the expression γij = ei · dej means that γij =

P

dγij = d (ei · dej ) , = dei ∧ dej , =

X

(ek · dei ) ∧ (ek · dej ) ,

k

=

X k

D.9. Here are two proofs:

γki ∧ γkj .

k

eki dekj , so:

173

Hints a. If we think of x and e as functions on ⌜E3 , then rg∗ x = x, rg∗ e = eg. Hence rg∗ dx = dx and rg∗ de = (de)g. So rg∗ ω = rg∗ (e> dx) = (eg)> dx = g> e> dx = g> ω and rg∗ γ = rg∗ (e> de) = (eg)> d(eg) = g> e> de g = g> γg. P b. The action is rg (x, e) = (x, eg), where (eg)i = j gji ej . Hence rg0 (x, e)(x, ˙ e) ˙ = (x, ˙

X

gji e˙ j ).

j

(rg∗ ω)(x,e) (x, ˙ e) ˙ = ωrg (x,e) rg0 (x, e)(x, ˙ e), ˙ = ω(x,eg) (x, ˙

X

gji e˙ j ),

j

= (eg) · x, ˙ =

X

gji ej · x, ˙

j

=

X

gji ω(x,e) (x, ˙ e). ˙

j

D.13. For any tangent vectors u, v ∈ Tx S, II(u, v) = II(v, u) =

X

aij ui vj e3 ,

ij

We can write II as II(u, v) =

P

aij ωi (u)ωj (v)e3 , i.e. II = ω> aωe3 , so

rg∗ II = rg∗ ω> aωe3 = (h> ω)> (nh> ah)(h> ω)(ne3 ) = ω> aωe3 = II, i.e. rg∗ II = II so II is invariant. D.16. Take any point p0 ∈ E3 . At a point x0 where the surface acheives maximal distance from some point p0 ∈ E3 , differentiate distance to see that Tx0 S is perpendicular to the ray from p0 to x0 . Translate x0 to the origin, and rescale and rotate to get p0 a unit vector above the origin. Apply our local picture of surfaces to see that S being inside the unit sphere around p0 forces S to be locally the graph of a function with critical zero at the origin, and eigenvalues of the second derivative both at least 1. D.17. Let ∂f ∂2f fi ..= , fij ..= . ∂xi ∂xi x2j Note that fj ∂xj e3 = pP . fk2 So

!

fj

de3 = d

pP

= −fj

fk2

∂xj ,

  fjk dxk 1 X 2 −3/2 fk 2fk fk` dx` ∂xj + pP ∂xj , 2 fk2

 = − e3

D2 f |df |

 e3 +

fjk dxk ∂xj . |df |

174

Hints

Note that ei · e3 = 0 for i = 1, 2, so

γi3 = ei · de3 =

ei

D2 f . |df |

E.1. Note that ρ rotates the point x0 of our plane around on a circle perpendicular to e1 , e3 and tangent to e2 , with radius r, so that dρ 0 x = re2 . dθ Therefore ω1 = e1 · dx, = e1 · d ρx0 ,



dρ 0 x dθ, dθ 0 = e1 · ρ e1 ds + e1 · re2 dθ,

= e1 · ρ dx0 + e1 ·

= e1 · e1 ds = ds. Similarly, ω3 = 0 and ω2 = e2 · dx, = e2 · d ρx0 ,



dρ 0 x dθ, dθ = e2 · ρ e01 + e2 · re2 dθ,

= e2 · ρ dx0 + e2 ·

= r dθ. Note that de01 = ϕ˙ e03 ds, de02 = 0, de03 = −ϕ˙ e01 ds, and dρ = ρ

0 1 0

−1 0 0

0 0 0

! dθ.

175

Hints

Compute de1 = d(ρe01 ), = dρ e01 + ρ de01 , =ρ

0 1 0

−1 0 0

0 0 0

0 1 0

−1 0 0

0 0 0

0 cos ϕ 0

! e01 dθ + ρϕe ˙ 03 ds,

!

cos ϕ 0 sin ϕ

! dθ + ϕe ˙ 3 ds,

! dθ + ϕe ˙ 3 ds,

= cos ϕe2 dθ + ϕe ˙ 3 ds. and similarly de2 = d(ρe02 ), = dρ e02 , −1 0 0

0 1 0

= −ρ

1 0 0

=−

cos θ sin θ 0

0 0 0

! e02 dθ

! dθ

! dθ.

Finally, de3 = d(ρe03 ), = dρ e03 + ρ de03 , =ρ

0 1 0

−1 0 0

0 0 0

0 1 0

−1 0 0

0 0 0

0 − sin ϕ 0

!

!

e03 dθ − ρϕe ˙ 01 ds, − sin ϕ 0 cos ϕ

! dθ − ϕe ˙ 1 ds,

= − sin ϕ e2 dθ − ϕe ˙ 1 ds.

! dθ − ϕe ˙ 1 ds,

176

Hints

Therefore γ12 = −γ21 , = −e2 · de1 , = −e2 · (cos ϕ e2 dθ + ϕe ˙ 3 ds), = − cos ϕ dθ. γ13 = e1 · de3 , = −ϕ˙ ds,

γ23 = e2 · de3 , = − sin ϕ dθ.

E.2. If sin ϕ = 0 everywhere, the curve is a horizontal line and the surface of revolution an annulus. Suppose that sin ϕ is not everywhere zero; pick an interval on which sin ϕ 6= 0, and let ε be the sign of sin ϕ. Differentiate to find that the function β ..= εr

p

1 − r˙ 2 + H0 r2

is constant along solutions, say equal to β0 . Solve for r: ˙ β0 − H0 r2 . r2

r˙ 2 = 1 − Substitute u = r2 : u˙ = 2

p

H0 u2 + u − β0 .

Integrate:

Z

du 2

p

= s.

H0 u2 + u − β0

This integral can be solved in elementary functions giving s = s(r); for example, if H0 > 0,

 √ p  (4β0 H0 + 1) log 2 H0 H0 u2 + u − β0 + 2H0 u + 1 s=

,

3/2

16H0

 √

(4β0 H0 + 1) log 2 H0

p

H0 r4 + r2 − β0 + 2H0 r2 + 1

=

 ,

3/2

16H0

The surfaces of revolution of constant positive mean curvature are obtained by solving implicitly for r = r(s). E.6. Taking real and imaginary parts q = q1 + iq2 :



a11 a12

a12 a22



 =

H + q1 −q2

−q2 H − q1

 .

177

Hints

So H = (a11 + a22 )/2 is the mean curvature, and q=

a11 − a22 − 2i a12 . 2

Note that |q|2 = q q¯ = H 2 − K, so q = 0 precisely at umbilic points. E.9. In our complex notation, we are identifying the matrix −1 0 0

0 1 0

0 0 0

!

with the complex number i. The fibers of ⌜S → S have the form (x(t), e(t)) = (x0 , e0 eit ). So if we write out γ = e> de in our complex notation, on each fiber, γ = i dt, so α = γ12 = −dt. We are given that f (x(t), e(t)) = f (x0 , e0 eit ) = eikt f (x0 , e0 ), so on the fiber

df = ikf, dt

i.e. df + ikf α = 0. So in complex notation, df + ikf α vanishes on vertical vectors for ⌜S → S, i.e. is a multiple of ω, ω ¯. E.10. Dq = a111 − 3a122 + i(a222 − 3a112 ), ¯ = 2(H1 − iH2 ). Dq

E.11.



d(γ13 − k1 ω1 ) d(γ23 − k2 ω2 )



−γ12 ∧ γ23 − dk1 ∧ ω1 + k1 γ12 ∧ ω2 −γ21 ∧ γ13 − dk2 ∧ ω2 + k2 γ21 ∧ ω1



−γ12 ∧ k2 ω2 − dk1 ∧ ω1 + k1 γ12 ∧ ω2 γ12 ∧ k1 ω1 − dk2 ∧ ω2 − k2 γ12 ∧ ω1

0= = =





dk1 ∧ ω1 + (k2 − k1 )γ12 ∧ ω2 =− (k2 − k1 )γ12 ∧ ω1 + dk2 ∧ ω2



E.12. Arrange e1 , e2 to diagonalize the shape operator, so γ13 = k1 ω1 , γ23 = k2 ω2 ,

 

178

Hints

with k1 constant. Differentiate to find that γ12 is a multiple of ω2 . So then along the flow of e1 , γ12 = 0 and γ23 = 0 i.e. e2 is constant, so e1 , e3 rotate in the plane, with γ13 = k1 , so at a constant rate, i.e. on a circle. E.13. Suppose that θ(x, y) is a smooth solution of the sine–Gordon equation θxy = sin θ, on an open subset U ⊂ R2 of the x, y plane. Take the exterior differential system generated by ω1 − sin(θ/2)d(y − x), ω2 − cos(θ/2)d(x + y), ω3 , 2γ12 + θx dx − θy dy, γ13 + cos(θ/2)d(y − x), γ23 − sin(θ/2)d(x + y) on U × ⌜E3 . We want to prove that this system is Frobenius. It is invariant under rigid motion, so its integral surfaces are permuted by rigid motion. We want to prove that its maximal integral surfaces are immersed into R3 by the obvious projection. Hence it determines a unique surface in Euclidean 3-space, up to rigid motion, with constant negative Gauss curvature equal to −1. To prove that the system is Frobenius, take exterior derivatives, modulo the 1forms written above, and plug in that our given θ satisfies the sine–Gordon equation. So the 8-dimensional manifold U ×⌜E3 is foliated by integral surfaces of these 6 linearly independent 1-forms, leaves of a foliation. Each integral surface has dx, dy linearly independent, and so has ω1 , ω2 linearly independent. So it projects by immersion to 3-dimensional Euclidean space. Proceed as in the solution of problem 4.1 on page 34. Careful: on a multiply connected domain U ⊂ R2, it is possible that a solution θ(x, y) of sine–Gordon might give rise to a surface which is a nontrivial covering of U . F.5. e3

  d cos θe02 + sin θe03 = e3 −θ˙ sin θe02 + cos θe˙0 2 + θ˙ cos θe03 + sin θe˙0 3 , ds  = e3 −θ˙ sin θe02 + cos θ(−ke1 + te03 ) + θ˙ cos θe03 − t sin θe02 , = e3 −θ˙ sin θe02 + t cos θe03 + θ˙ cos θe03 − t sin θe02 ,



= θ˙ sin2 θt cos2 θθ˙ cos2 θ + t sin2 θ, = θ˙ + t.

F.6. We get e˙ 1 = 0 along such a curve, at that point, so 0 = γ21 = γ31 , so a11 = 0. Since this occurs in all directions, a = 0. G.4. The fact that exp0x0 = I means that at the origin, ω12 + ω22 = dx2 + dy 2 , in rectangular coordinates, i.e. near the origin ω12 + ω22 − dr2 − r2 dθ2

179

Hints

is smooth. Expand out, and plug in r dθ = cos θ dy − sin θ dx to see that ω12

+

ω22

2

2

2

= dx + dy +

 −1

 2



= dr + r dθ + 2

 2 h r

2

h r

r2 dθ2 ,

− 1 (cos θ dy − sin θ dx) .

In particular, h →1 r as r → 0. Hence h → 0 as r → 0 and ∂r h → lim

r→0

h = 1. r

G.7. Gauss–Bonnet G.8. If our geodesic is x(t), then f (x(t)) has local minima as its only critical points. Moreover, they are isolated, as f (x(t) is critical only where x(t) reaches the minimum of f or becomes tangent to a level set of f . If f (x(t) stays bounded as t → ∞, then x(t) stays in a compact sublevel set of f . But f (x(t)) increases for large enough t, so x(t) approaches the boundary of that sublevel set, i.e. a level set. Geodesics near each level set enter or leave soon, a contradiction. G.10. The differential equation forces h = r − Kr3 /3! + O(r)4 , and inductively forces ∂rk ∂θ` h = O(r)` . Note that sin θ ∂θ , r cos θ ∂y = sin θ∂r + ∂θ . r

∂x = cos θ∂r −

So h2 − r2 is a smooth function of x, y at the origin. dr2 + h2 dθ2 = dr2 + r2 dθ2 + (h2 − r2 )dθ2 , (h2 − r2 )(−x dy + y dx) , r2 K(x2 + y 2 ) + O(r)3 = dx2 + dy 2 − (−x dy + y dx) 3 = dr2 + r2 dθ2 +

is smooth at the origin.

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183

Index

absorbing torsion, 17 adapted frame, 135 frame bundle, 34 tableau, 16 almost complex structure, 26 analytic, 1 convergence, 97 function, 78 area form, 117 asymptotic curve, 67 asymptotically simple, 97

vector Cauchy, 62 Chern Shiing-Shen, 147 circle, 141 codimension, 1 coframe, 8 coframing, 8 compatibility, 7 complete surface, 122, 140 vector field, 60, 122 complex differential, 119 conformal, 69 connection forms Levi-Civita, 108 contact, 35 convergence analytic, 97 germ, 98 of differential forms, 60 coordinates isothermal, 71 curvature Gauss, 111 geodesic, 125 principal, 112 vector, 104

Bonnet theorem, 117 borrow polars, 18 Bryant Robert, 147 bundle frame, 103 bundled, 61 Cartan Élie, 146 Henri, 98 lemma, 6 test, 40 Cartan–Kähler theorem, 4, 54 Cauchy Augustin Louis, 146 characteristic, 62, 63 vector, 62 Cauchy–Kovalevskaya theorem, 80, 87, 92 Cauchy–Riemann equations, 25 character, 3, 14 characteristic, 48 Cauchy, 63 variety, 48, 90 nonlinear differential equation, 92

Darboux frame, 120 degree, 130 determined, 92 integral element, 49 element integral, 2 elementary graded ring, 97 Euler characteristic, 128, 129 Euler–Tricomi equation, 94 exponential map, 136

185

186

Index

exterior differential system, 59 pullback, 63 pushforward, 63 fiber, 35 finite type, 61 flow, geodesic, 133 form soldering, 107 formal solution, 79 Taylor series, 77 frame, 106 adapted, 125 bundle, 107 Darboux, 120 orthonormal, 101, 144 frame bundle, 103 adapted, 34 Frobenius, 6 Frobenius theorem, 6, 7, 9, 49, 60, 63, 168 fundamental form second, 110 Gauss curvature, 111 Gauss map, 117, 130 Gauss–Bonnet theorem, 129, 145 generic, 39 geodesic, 134, 141 curvature, 125 flow, 133 minimal, 141 spray, 133 germ, 98 convergence, 98 grade, 41 graded module, 97 asymptotically simple, 97 elementary, 97 ring, 97 elementary, 97 Grassmann bundle, 3 Griffiths Phillip, 148 helix, 105 Hilbert

basis theorem, 98 surface theorem, 122 holomorphic, 25 curve, 31 differential, 120 Hopf differential, 117 Hopf–Rinow theorem, 141 hyperplane, 1 hypersurface, 1 index of vector field, 128 injectivity radius, 137 integral element, 2 ordinary, 39 regular, 39 flag, 3 line, 2 manifold, 1 plane, 2 involutive, 3 isothermal coordinates, 71 Jacobi identity, 22 Kähler Eric, 146 Korn–Lichtenstein theorem, 120 Kovalevskaya Sofia Vasilyevna, 146 Krull intersection theorem, 98 Kuranishi Masatsugu, 147 Lagrangian submanifold, 15 special, 4 Laplace equation, 94 operator, 94 lemma Cartan’s, 6 Phat Nguyen, 97 length space, 141 Levi-Civita connection, 9 forms, 108 Lie bracket, 7 group, 21 third theorem, 21

187

Index

Lie’s third theorem, 51 Liebmann, H., 121 line integral, 2 linearization, 47, 91 local deformation, 44 locally generated, 6 locally unique, 54 majorize, 77 manifold integral, 1 restraining, 53, 54 Maurer–Cartan form, 21, 30 mean curvature, 111 vector, 111 minimal geodesic, 141 minimal surface equation, 94 Nijenhuis tensor, 156 Noetherian, 98 nondegenerate ribbon, 67 orange, 68 ordinary integral element, 39 orthonormal frame, 101, 144 parallel surfaces, 118 peel, 68 Phat Nguyen lemma, 97 plane integral, 2 polar, 14 polar equation, 2 predicted dimension, 3 principal curvatures, 112 directions, 112 projective space, 89 prolongation, 35 pullback, 6 exterior differential system, 63 pushforward exterior differential system, 63 radius injectivity, 137 regular

integral element, 39 restraining manifold, 53, 54 retracting space, 63 ribbon, 67 nondegenerate, 67 Riemannian metric, 143 ruling line, 67 Sard’s theorem, 130 second fundamental form, 110 semipositive, 41 Serret–Frenet frame, 105 shape operator, 110 sign of map between manifolds of equal dimension, 129 sine–Gordon equation, 123 soldering forms, 9, 107 solution formal, 79 special Lagrangian submanifold, 4 spray geodesic, 133 structure equations, 109, 111 of Euclidean space, 108 submanifold, 1 special Lagrangian, 4 surface complete, 122, 140 of revolution, 115 principal curvatures, 112 principal directions, 112 Weingarten, 73 symbol, 88 symmetry vector field, 59 tableau, 14 Taylor series formal, 77 test Cartan’s, 40 theorem Bonnet, 117 Cartan–Kähler, 4, 54 Cauchy–Kovalevskaya, 80, 87, 92 Frobenius, 6, 7, 9, 49, 63, 168 Gauss–Bonnet, 129, 145 Hilbert, 122 Hilbert basis, 98 Hopf–Rinow, 141

188

Index

Korn–Lichtenstein, 120 Krull intersection, 98 Lie’s third, 21, 51 Sard, 130 theorema egregium, 142, 145 third fundamental form, 112 torsion, 36 absorbing, 17 of a curve, 104 tensor, 156 triply orthogonal web, 8 umbilic, 8, 112 variety characteristic, 48 vector bundle, 62, 150 vector field complete, 60, 122 Weingarten surface, 73